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✓ 19 results · all verified · 15 also independently AI-judged

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hp finite elements and spectral elements

1 · Prerequisites

2 · Summary

This page defines variable-degree spaces on matching affine simplicial and tensor-box meshes. It proves the p-explicit simplex inverse trace estimate, an ordinary Sobolev triangle projector, a degree-uniform strong-edge lift, and the resulting varying-degree conforming interpolant with its actual lowest shared-edge degree dependence. It also proves analytic fixed-mesh box p convergence, GLL p-uniform norm equivalence with the exact Strang defect retained, weighted pointwise corner bounds, and the full two-dimensional geometric hp energy rate. The corner proof uses its direct weighted and edge-lift suppliers and does not assume H2 regularity on singular caps. The local estimates concern affine cells. The declared same-frontier curved-mesh prerequisite supplies a separate geometric framework with its own mapping and transfer hypotheses; affine approximation and quadrature claims here do not assert curved-cell consistency.

The diagram records the trace-matching convention for two cells with unequal local degrees. The shared trace lies in the lower-degree face space; geometric adjacency alone does not impose conformity.

\begin{tikzpicture}[scale=0.9]
  \draw[thick] (0,0) rectangle (2,1.4);
  \draw[thick] (2,0) rectangle (4,1.4);
  \draw[dashed] (2,0) -- (2,1.4);
  \node at (1,0.75) {$K_1,\ p_1=3$};
  \node at (3,0.75) {$K_2,\ p_2=5$};
  \node[rotate=90] at (2.28,0.7) {$p_F=\min(p_1,p_2)$};
  \draw[<->] (0,-0.25) -- (4,-0.25) node[midway,below] {$h$-refinement changes cell sizes};
\end{tikzpicture}

For a fixed tensor mesh, the integrated Legendre projector gives an explicit coefficient-based p estimate, and a reference-cell analytic derivative bound yields exponential p convergence. For polygonal corner singularities, the proved 2D branch uses nested matching shape-regular geometric corner meshes, at most $O (k)$ triangles, and uniform total degree $p_{k} ≃ c k$ ; it yields an $e^{- b N^{1 / 3}}$ best-approximation and Galerkin energy rate for the stated weighted-analytic class. The general varying-degree Galerkin theorem now carries the local triangle comparison estimates through Céa, retaining each cell diameter, the minimum incident shared-edge degree, the Sobolev regularity and the actual continuity/coercivity constants. For arbitrary degrees at least one, piecewise H2 data already give an explicit first-order h energy estimate; more generally integer regularity $s \geq 2$ gives $h^{s - 1}$ when the effective trace degree is at least $s - 1$ . For cellwise integer regularity $r \geq 5$ , the additional p-dependent bound has $h^{r - 1}$ times the proved factorial factor, bounded at sufficiently high trace degree $m$ by $(4 / m)^{r - 4}$ . Both the H1 norm and zero-Dirichlet gradient norm are covered. No optimal p rate or growth estimate for the $r$ -dependent constant is asserted.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

Endpoint-preserving integrated Legendre projector

Definition

For $p \geq 1$ and $g \in L^{2} (- 1, 1)$ , define the degree- $(p - 1)$ Legendre orthogonal projection

Π_{p - 1} g : = \sum_{n = 0}^{p - 1} \frac{2 n + 1}{2} (\int_{- 1}^{1} g (t) P_{n} (t) d t) P_{n} .

For $c \in R$ , define the endpoint-anchored integrated projector

I_{p} (c, g) (x) : = c + \int_{- 1}^{x} Π_{p - 1} g (t) d t, - 1 \leq x \leq 1.

If an absolutely continuous function $f$ is represented as $f (x) = f (- 1) + \int_{- 1}^{x} f^{'} (t) d t$ with $f^{'} \in L^{2}$ , write $I_{p} f : = I_{p} (f (- 1), f^{'})$ . The finite sum and integral are defined for every such input; no choice axiom is used in this definition.

DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

Variable-degree conforming hp spaces on matching affine meshes

Definition

Let $d \geq 2$ and let $T$ be a finite matching affine simplicial mesh of a bounded polyhedral open set $Ω \subset R^{d}$ in the sense of Matching affine simplicial meshes and shape regularity. A degree map is a function $p : T \to N_{0}$ with finite values. The broken simplex space associated with $(T, p)$ is

V^{b r} (T, p) : = {v \in L^{2} (Ω) : v ∣_{K} \in P_{p_{K}} (K) for every K \in T},

where $p_{K} : = p (K)$ and $P_{m} (K)$ denotes restrictions of real polynomials of total degree at most $m$ . Its conforming hp space is the subspace of functions whose two polynomial traces agree on every interior facet. The zero-boundary hp space further requires the polynomial trace to vanish on every boundary facet. For an interior facet $F = K \cap K^{'}$ , its trace degree is $p_{F} : = \min (p_{K}, p_{K^{'}})$ ; equality of the two traces means that their common polynomial belongs to $P_{p_{F}} (F)$ .

A matching affine tensor-box mesh is a finite face-to-face partition of a bounded polytope into affine images of $[- 1, 1]^{d}$ , with pairwise disjoint interiors. For a scalar degree $p_{K}$ on each box, the local tensor space is $Q_{p_{K}} (K)$ , the pullback of polynomials of degree at most $p_{K}$ in each reference coordinate. The conforming box hp space is defined by equality of polynomial traces across every shared face, and its zero-boundary subspace by zero traces on the exterior faces. For a shared face $F = K \cap K^{'}$ , set $p_{F} = \min (p_{K}, p_{K^{'}})$ ; its common trace lies in $Q_{p_{F}} (F)$ in tangential reference coordinates. This definition of tensor boxes does not include curved or non-affine cells.

On either mesh, h-refinement replaces one cell by a finite matching partition of that cell and assigns degrees to the children; p-refinement raises at least one cell degree while leaving the mesh fixed; hp-refinement performs both operations. A local polynomial basis is hierarchical if its ordered degree layers span the nested spaces $P_{0} \subset \dots \subset P_{p}$ (or $Q_{0} \subset \dots \subset Q_{p}$ ), with each new layer extending the preceding span. For example, pullbacks of reference monomials ${\hat{x}}^{ν}$ ordered by $∣ ν ∣$ give such a basis for $P_{p}$ , and tensor monomials $ξ^{ν}$ ordered by $\max_{i} ν_{i}$ give one for $Q_{p}$ . This basis convention does not assert stability of a particular basis or projector.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

Legendre-Gauss-Lobatto nodes are distinct and interior critical points are simple

Statement

For every integer $p \geq 1$ , the Legendre polynomial $P_{p}$ has exactly $p$ simple roots in $(- 1, 1)$ , and $P_{p}^{'}$ has exactly $p - 1$ simple roots in $(- 1, 1)$ , one between each consecutive pair of roots of $P_{p}$ . Consequently, the $p + 1$ nodes consisting of $- 1$ , $1$ , and the roots of $P_{p}^{'}$ are pairwise distinct.

Facts & Assumptions

Given: An integer $p \geq 1$ and the Legendre polynomials $P_{n}$ .

[F1]

Legendre polynomials by Rodrigues' formula and their unweighted pairing: $P_{p}$ is a nonzero degree- $p$ polynomial.

[F2]

Legendre polynomials have explicit norms and weighted derivative orthogonality: $P_{p}$ is orthogonal in unweighted $L^{2} (- 1, 1)$ to every polynomial of degree less than $p$ .

[F3]

Rolle's theorem: if $a < b$ , $f$ is continuous on $[a, b]$ , differentiable at every point of $(a, b)$ , and $f (a) = f (b)$ , then $f^{'} (c) = 0$ for some $c \in (a, b)$ : between two distinct zeros of a differentiable real function there is a zero of its derivative.

Proof

1.1F1F2givenalgebra

Suppose $P_{p}$ has only $m < p$ sign-changing roots $r_{1}, \dots, r_{m}$ in $(- 1, 1)$ and set $q (x) = \prod_{j = 1}^{m} (x - r_{j})$ , with $q = 1$ if $m = 0$ . At each sign-changing root both $P_{p}$ and $q$ change sign, while at every other real root $P_{p}$ does not change sign; therefore $P_{p} q$ has one constant sign on $(- 1, 1)$ and is not identically zero. It is continuous and nonzero on some open interval, so its integral is nonzero. But $\deg q = m < p$ , and [F2] gives $\int_{- 1}^{1} P_{p} q = 0$ , a contradiction. Thus $P_{p}$ has at least $p$ sign-changing roots in the open interval.

2.1F1F3step 1.1givenalgebra∎

A degree- $p$ polynomial cannot have more than $p$ distinct roots. The $p$ roots obtained in step 1.1 therefore exhaust its degree; each has multiplicity one and there are no other roots. Rolle's theorem [F3] between consecutive roots gives $p - 1$ distinct roots of $P_{p}^{'}$ in $(- 1, 1)$ . Since $P_{p}^{'}$ has degree $p - 1$ , these are all its roots and each is simple. None equals an endpoint, so adjoining $- 1$ and $1$ gives $p + 1$ distinct nodes, including the case $p = 1$ where there are no interior critical points.

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Weighted Jacobi endpoint kernel on the reference interval

Statement

For an integer $d \geq 1$ , put $w_{d} (t) = (1 - t)^{d - 1}$ on $[0, 1]$ and, for $n \geq 0$ , define $ϕ_{n} (t) : = \frac{(- 1)^{n}}{n!} (1 - t)^{- (d - 1)} \frac{d^{n}}{d t^{n}} (t^{n} (1 - t)^{n + d - 1}) .$ The quotient extends to a polynomial of degree $n$ . These polynomials are pairwise orthogonal in $L^{2} (w_{d} d t)$ , with $ϕ_{n} (0) = (- 1)^{n}, lc (ϕ_{n}) = (\binom{2 n + d - 1}{n}), \int_{0}^{1} w_{d} ϕ_{n}^{2} d t = \frac{1}{2 n + d} .$ Consequently, for every $p \geq 0$ the squared norm of endpoint evaluation $q \mapsto q (0)$ on $P_{p} ([0, 1])$ with this weighted $L^{2}$ norm is exactly $(p + 1) (p + d)$ .

Facts & Assumptions

Given: Integers $d \geq 1$ and $p \geq 0$ , the weight $w_{d} (t) = (1 - t)^{d - 1}$ , and polynomials on $[0, 1]$ .

[A1]

The factor $t^{n} (1 - t)^{n + d - 1}$ and its derivatives of orders $0, \dots, n - 1$ vanish at both endpoints; integration by parts therefore has no boundary terms.

Proof

1.1givenA1algebra

Apply Leibniz's rule to the $n$ th derivative in the definition. Every term differentiates $(1 - t)^{n + d - 1}$ at most $n$ times, so it contains a factor $(1 - t)^{d - 1}$ . After division the result is a polynomial. The original factor has degree $2 n + d - 1$ and leading coefficient $(- 1)^{n + d - 1}$ ; differentiating $n$ times and dividing by the leading term $(- 1)^{d - 1} t^{d - 1}$ of $(1 - t)^{d - 1}$ shows that $ϕ_{n}$ has degree $n$ and leading coefficient $(\binom{2 n + d - 1}{n})$ . The coefficient of $t^{n}$ in $t^{n} (1 - t)^{n + d - 1}$ is $1$ , so its $n$ th derivative at zero is $n!$ and $ϕ_{n} (0) = (- 1)^{n}$ .

2.1givenA1step 1.1algebra

If $r$ has degree below $n$ , multiply the Rodrigues formula by $w_{d} r$ and integrate. Integrating by parts $n$ times is allowed by [A1] and gives $\int_{0}^{1} w_{d} (t) ϕ_{n} (t) r (t) d t = \frac{1}{n!} \int_{0}^{1} t^{n} (1 - t)^{n + d - 1} r^{(n)} (t) d t = 0$ , because $r^{(n)} = 0$ . Thus $ϕ_{n}$ is orthogonal to every lower-degree polynomial, and the degree conclusion in step 1.1 makes the family an orthogonal basis of each polynomial space.

3.1givenA1step 2.1algebra

For nonnegative integers $a, b$ , let $B (a, b) = \int_{0}^{1} t^{a} (1 - t)^{b} d t$ . The case $a = 0$ gives $B (0, b) = 1 / (b + 1)$ ; for $a \geq 1$ , integration by parts with $u = t^{a}$ and $d v = (1 - t)^{b} d t$ gives $B (a, b) = a B (a - 1, b + 1) / (b + 1)$ . Iterating yields $B (a, b) = a! b! / (a + b + 1)!$ . Applying $n$ integrations by parts to the Rodrigues formula with $r (t) = t^{n}$ gives $\int_{0}^{1} w_{d} ϕ_{n} t^{n} d t = B (n, n + d - 1) = n! (n + d - 1)! / (2 n + d)!$ . Since $t^{n} = ϕ_{n} / κ_{n} + r$ for some $r$ of degree below $n$ , where $κ_{n} = (\binom{2 n + d - 1}{n})$ , orthogonality gives $∥ ϕ_{n} ∥_{L^{2} (w_{d})}^{2} = κ_{n} B (n, n + d - 1) = 1 / (2 n + d)$ .

4.1step 1.1step 2.1step 3.1algebra∎

Write $q = \sum_{n = 0}^{p} c_{n} ϕ_{n}$ . By the endpoint and norm identities, Cauchy–Schwarz gives $∣ q (0) ∣^{2} = {∣ \sum_{n = 0}^{p} c_{n} (- 1)^{n} ∣}^{2} \leq (\sum_{n = 0}^{p} (2 n + d)) (\sum_{n = 0}^{p} \frac{∣ c_{n} ∣^{2}}{2 n + d}) = (p + 1) (p + d) ∥ q ∥_{L^{2} (w_{d})}^{2}$ . Equality is attained by the polynomial $K_{p} (t, 0) = \sum_{n = 0}^{p} (2 n + d) (- 1)^{n} ϕ_{n} (t)$ : its endpoint value and squared norm both equal $\sum_{n = 0}^{p} (2 n + d) = (p + 1) (p + d)$ . Hence the evaluation norm squared is exactly the stated value.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

Weighted simplex polynomial projection has explicit p-rates

Statement

Let $\hat{K}$ be the reference simplex of Matching affine simplicial meshes and shape regularity, let $α = (α_{0}, \dots, α_{d})$ have nonnegative real entries, and let $ω_{α} = \prod_{j = 0}^{d} λ_{j}^{α_{j}}$ , where $λ_{j}$ are its barycentric coordinates. Give polynomials the inner product $⟨ f, g ⟩_{α} = \int_{\hat{K}} ω_{α} f g$ . Put $P_{- 1} = {0}$ and let $E_{n} = P_{n} (\hat{K}) ⊖ P_{n - 1} (\hat{K})$ in this weighted inner product. For a finite sum $u = \sum_{n \geq 0} u_{n}$ , $u_{n} \in E_{n}$ , define

∥ u ∥_{H_{α}^{s}}^{2} : = \sum_{n \geq 0} (1 + μ_{n})^{s} ∥ u_{n} ∥_{L^{2} (ω_{α})}^{2}, μ_{n} = n (n + d + ∣ α ∣),

and define $H_{α}^{s}$ as the completion of these finite sums. If $m \geq ℓ \geq 0$ , its orthogonal degree truncation $P_{p} u = \sum_{n = 0}^{p} u_{n}$ obeys

∥ u - P_{p} u ∥_{H_{α}^{ℓ}} \leq (1 + μ_{p + 1})^{- (m - ℓ) / 2} ∥ u ∥_{H_{α}^{m}} .

For $q \in P_{p} (\hat{K})$ , $∥ q ∥_{H_{α}^{m}} \leq (1 + μ_{p})^{(m - ℓ) / 2} ∥ q ∥_{H_{α}^{ℓ}}$ . These are estimates in the explicitly coefficient-defined scale $H_{α}^{s}$ ; they do not identify that scale with ordinary unweighted $H^{s} (\hat{K})$ .

Facts & Assumptions

Given: A dimension $d \geq 2$ , $α_{j} \geq 0$ , real exponents $m \geq ℓ \geq 0$ , and degree $p \geq 0$ .

[F1]

The reference simplex is a bounded nondegenerate simplex with barycentric coordinates; its matching-mesh and reference-map conventions are fixed by Matching affine simplicial meshes and shape regularity.

[A1]

$P_{n} (\hat{K})$ is finite-dimensional, and $ω_{α}$ is positive in the open simplex and integrable. Thus its weighted $L^{2}$ inner product is positive definite on polynomials.

Proof

1.1F1A1givenalgebra

Since $ω_{α} > 0$ in the interior, a nonzero polynomial cannot have weighted square integral zero: continuity gives a ball on which its absolute value is bounded below by a positive number. Integrability follows from $α_{j} \geq 0$ on the bounded simplex. Hence the weighted inner product is positive definite on each finite-dimensional polynomial space. Orthogonal projection onto $P_{n - 1}$ therefore gives the finite-dimensional orthogonal decomposition $P_{n} = E_{0} \oplus \dots \oplus E_{n}$ . The scale in the Statement is the completion of this orthogonal-sum norm, so its coefficient description and truncation are well-defined.

2.1givenstep 1.1algebra

For a finite sum $u = \sum u_{n}$ , orthogonality gives $∥ u - P_{p} u ∥_{H_{α}^{ℓ}}^{2} = \sum_{n > p} (1 + μ_{n})^{ℓ} ∥ u_{n} ∥_{α}^{2} \leq (1 + μ_{p + 1})^{ℓ - m} \sum_{n > p} (1 + μ_{n})^{m} ∥ u_{n} ∥_{α}^{2}$ , because $μ_{n} = n (n + d + ∣ α ∣)$ is increasing for $n \geq 0$ . Taking square roots proves the tail estimate for finite sums.

3.1step 1.1step 2.1algebra∎

Truncation is contractive in every coefficient norm $H_{α}^{s}$ , and finite sums are dense by the definition of the completion. The finite-sum estimate in step 2.1 therefore passes to every $u \in H_{α}^{m}$ . If $q \in P_{p}$ , each degree layer has $n \leq p$ , so $(1 + μ_{n})^{m - ℓ} \leq (1 + μ_{p})^{m - ℓ}$ term by term; summing proves the inverse estimate. Neither argument invokes a derivative norm, which is why neither implies an unweighted Sobolev estimate.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

A degree-uniform polynomial edge lift with the strong edge norm

Statement

Assume Countable Choice. Let $p \in N_{0}$ , let $T = {(x, y) : x > 0, y > 0, x + y < 1}$ , and let $q$ be a real polynomial of degree at most $p$ satisfying $q (0) = q (1) = 0$ . Define the polynomial $R (t) = q (t) / t$ by cancellation at $t = 0$ , and put

L q (x, y) = q (x) - x R (1 - y) .

Then $L q \in P_{p} (T)$ , it equals $q$ on $y = 0$ , and it vanishes on $x = 0$ and $x + y = 1$ . Moreover,

∥ L q ∥_{H^{1} (T)} \leq 4 ∥ q^{'} ∥_{L^{2} (0, 1)},

independently of $p$ . Permuting the three vertices gives the same bound, up to one fixed constant, for each reference edge. If $K$ is a shape-regular affine triangle, $E$ is an edge of length $ℓ_{E}$ , and $h_{K} = diam (K)$ , the transported lift satisfies

∣ L_{E} q ∣_{H^{1} (K)} \leq C_{γ} ℓ_{E}^{1 / 2} ∥ \partial_{τ} q ∥_{L^{2} (E)}, ∥ L_{E} q ∥_{L^{2} (K)} \leq C_{γ} h_{K} ℓ_{E}^{1 / 2} ∥ \partial_{τ} q ∥_{L^{2} (E)} .

Consequently continuous polynomial data on the three edges of a triangle, each of degree at most the cell degree, have a polynomial extension of that cell degree. The estimate uses the strong $H^{1}$ edge norm and asserts no optimal $H^{1 / 2}$ trace-lifting bound.

Facts & Assumptions

Given: Countable Choice; the reference triangle; a polynomial $q$ with the stated endpoint values; and, for the physical-cell assertion, a shape-regular affine triangle and one of its edges.

[F1]

A shape-regular affine triangle is a nondegenerate affine simplex with diameter-to-inradius ratio bounded by a fixed constant; its affine map has singular values, determinant, edge Jacobian, and altitude controlled by the cell diameter. (Matching affine simplicial meshes and shape regularity, Shape regularity controls affine simplex geometry)

[F2]

The Sobolev $H^{1}$ norm is the $L^{2}$ norm of the function and its weak first derivatives. (Integer-order Sobolev spaces and their norms)

[F3]

Fubini applies to the finite integrable polynomial expressions on the reference triangle, and Cauchy–Schwarz bounds the one-dimensional polynomial integral. (Fubini's theorem for L^1 functions on a sigma-finite product, Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs)

[A1]

Countable Choice is the exact assumption used by [F1]; no arbitrary family of polynomial lifts is selected. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

Proof

1.1givenconstructalgebra

Since $q (0) = 0$ , polynomial division gives a polynomial $R$ of degree at most $p - 1$ when $p \geq 1$ ; for $p = 0$ or $p = 1$ the endpoint conditions force $q = 0$ . Thus $L q$ has total degree at most $p$ . At $y = 0$ , $R (1) = q (1) = 0$ , so $L q = q (x)$ ; at $x = 0$ it is zero; and on $x + y = 1$ one has $1 - y = x$ , so $x R (1 - y) = x R (x) = q (x)$ .

2.1F2F3step 1.1algebra

Put $A = \int_{0}^{1} ∣ q^{'} (t) ∣^{2} d t$ . Polynomial integration term by term gives $q (t) = \int_{0}^{t} q^{'} (s) d s$ , so Cauchy–Schwarz [F3] yields $t ∣ R (t) ∣^{2} \leq \int_{0}^{t} ∣ q^{'} (s) ∣^{2} d s \leq A$ and hence $\int_{0}^{1} t ∣ R (t) ∣^{2} d t \leq A$ . The identity $t R^{'} (t) = q^{'} (t) - R (t)$ gives $\int_{0}^{1} t^{3} ∣ R^{'} (t) ∣^{2} d t \leq 2 \int_{0}^{1} t ∣ q^{'} ∣^{2} + 2 \int_{0}^{1} t ∣ R ∣^{2} \leq 4 A$ . Also $∣ q (t) ∣^{2} \leq t A$ , so $\int_{0}^{1} ∣ q ∣^{2} \leq A$ . Writing $t = 1 - y$ , the slices of $T$ are $0 < x < t$ . Therefore $∥ \partial_{x} L q ∥_{2}^{2} \leq 2 \int_{0}^{1} (1 - x) ∣ q^{'} (x) ∣^{2} d x + 2 \int_{0}^{1} t ∣ R (t) ∣^{2} d t \leq 4 A,$ $∥ \partial_{y} L q ∥_{2}^{2} = \frac{1}{3} \int_{0}^{1} t^{3} ∣ R^{'} (t) ∣^{2} d t \leq \frac{4}{3} A,$ and $∥ L q ∥_{2}^{2} \leq 2 \int_{0}^{1} (1 - x) ∣ q (x) ∣^{2} d x + \frac{2}{3} \int_{0}^{1} t^{3} ∣ R (t) ∣^{2} d t \leq \frac{8}{3} A .$ Fubini [F3] justifies the slice integrals. Using the Sobolev norm definition [F2], adding the three inequalities gives $∥ L q ∥_{H^{1} (T)}^{2} \leq 8 A \leq 16 A$ , proving the safe constant four.

3.1A1F1step 2.1algebra

Permute the reference vertices to obtain a lift on any selected edge; there are only six such affine maps, so the maximum of their fixed norm factors is independent of $p$ . Transport the reference estimate by the shape-regular affine map. The gradient change of variables is bounded by [F1], and the reference edge derivative norm is exactly scaled by $ℓ_{E}^{1 / 2} ∥ \partial_{τ} q ∥_{L^{2} (E)}$ . The physical $L^{2}$ norm has one additional factor bounded by $C_{γ} h_{K}$ by [F1]. This gives both stated estimates.

4.1step 1.1step 3.1constructalgebra∎

Let $g$ be continuous polynomial data on the three edges, with edge degrees at most $p$ . For $p = 0$ , continuity makes $g$ one constant and that constant extends it. For $p \geq 1$ , let $a$ be the affine interpolant of the three vertex values. Each edge residual $g ∣_{E} - a ∣_{E}$ vanishes at both endpoints and has degree at most $p$ . Lift the three residuals separately by step 1.1; each lift vanishes on the other two edges. Their sum plus $a$ therefore has trace exactly $g$ and degree at most $p$ . This works when edge degrees differ and requires no equal-degree assumption.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

An ordinary derivative controls the Legendre tail with a factorial ratio

Statement

Assume the Axiom of Choice for the $k \geq 1$ branch. Let $P_{n}$ be the real Legendre polynomial normalized by $P_{n} (1) = 1$ , and let $Π_{N}$ be the $L^{2} (- 1, 1; R)$ orthogonal projection onto $P_{N} (- 1, 1)$ . For every integer $k \geq 1$ , every integer $N \geq k - 1$ , and every real-valued $f \in H^{k} (- 1, 1)$ ,

∥ f - Π_{N} f ∥_{L^{2} (- 1, 1)} \leq {(\frac{(N + 1 - k)!}{(N + 1 + k)!})}^{1 / 2} ∥ f^{(k)} ∥_{L^{2} (- 1, 1)} .

For $k = 0$ , every integer $N \geq 0$ , and $f \in L^{2} (- 1, 1)$ , $∥ f - Π_{N} f ∥_{2} \leq ∥ f ∥_{2}$ . For $n, m \geq k$ the repeated derivative orthogonality identity is

\int_{- 1}^{1} (1 - x^{2})^{k} P_{n}^{(k)} (x) P_{m}^{(k)} (x) d x = \frac{2}{2 n + 1} \frac{(n + k)!}{(n - k)!} δ_{n m} .

Facts & Assumptions

Given: The Axiom of Choice; integers $k, N$ in the stated ranges; and a real-valued Sobolev class $f \in H^{k} (- 1, 1)$ .

[A1]

The Axiom of Choice is assumed only for the $k \geq 1$ branch. (The Axiom of Choice)

[F1]

The Rodrigues Legendre polynomials are mutually orthogonal in $L^{2} (- 1, 1)$ , have norm squared $2 / (2 n + 1)$ , and have degree $n$ with nonzero leading coefficient. (Legendre polynomials have explicit norms and weighted derivative orthogonality)

[F2]

Polynomials are uniformly dense in the continuous real functions on $[- 1, 1]$ (Polynomials are uniformly dense in $C ([a, b], R)$ for every closed interval), and smooth compactly supported functions are dense in $L^{2} (- 1, 1)$ (Smooth compactly supported functions of an open set are dense in $L^{2}$ ). Thus polynomials are dense in $L^{2} (- 1, 1)$ .

[F3]

Every one-dimensional $W^{1, 2}$ class on a bounded interval has an absolutely continuous representative satisfying $v (y) - v (x) = \int_{x}^{y} D v$ ; this theorem assumes Full Choice. Apply it to $f, f^{'}, \dots, f^{(k - 1)}$ . (One-dimensional $W^{1, p}$ functions have unique absolutely continuous representatives)

[F4]

Cauchy–Schwarz makes each displayed polynomial pairing finite. Expanding the norm after subtracting a finite orthogonal projection gives Bessel's inequality without any infinite-dimensional projection theorem. (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs)

Proof

1.1F1givenalgebra

Fix $k \geq 1$ and put $w_{k} (x) = (1 - x^{2})^{k}$ . For $n \geq k$ , the polynomial $Q_{n, k} : = D^{k} (w_{k} D^{k} P_{n})$ has degree $n$ . Integrating by parts $k$ times moves the outer derivative block onto a polynomial $q$ and gives $\int Q_{n, k} q = (- 1)^{k} \int w_{k} D^{k} P_{n} D^{k} q = \int P_{n} D^{k} (w_{k} D^{k} q)$ ; all boundary terms vanish because derivatives of $w_{k}$ of orders below $k$ vanish at both endpoints. The final polynomial paired with $P_{n}$ has degree at most $\deg q < n$ , so [F1] makes the integral zero. The leading coefficient of $Q_{n, k}$ is $(- 1)^{k} (n + k)! / (n - k)!$ times the leading coefficient of $P_{n}$ . Orthogonality and the degree statement in [F1] therefore give $Q_{n, k} = (- 1)^{k} λ_{n, k} P_{n}$ , where $λ_{n, k} = (n + k)! / (n - k)!$ .

2.1F1step 1.1algebra

For $n, m \geq k$ , integrate by parts $k$ times and use step 1.1 to obtain $\int_{- 1}^{1} w_{k} P_{n}^{(k)} P_{m}^{(k)} = (- 1)^{k} \int_{- 1}^{1} P_{n} D^{k} (w_{k} P_{m}^{(k)}) = λ_{m, k} \int_{- 1}^{1} P_{n} P_{m} .$ The first identity in [F1] makes this zero for $n \neq m$ and equal to $λ_{n, k} 2 / (2 n + 1)$ for $n = m$ , proving the repeated derivative orthogonality formula.

3.1A1F3F4step 1.1step 2.1algebra

Write $c_{n} = (f, P_{n})_{L^{2}} / ∥ P_{n} ∥_{2}^{2}$ . The absolutely continuous representatives in [F3] justify $k$ integrations by parts; all boundary terms vanish because $w_{k}$ has endpoint zeros of order $k$ . Step 1.1 then gives $\int_{- 1}^{1} w_{k} f^{(k)} P_{n}^{(k)} = λ_{n, k} c_{n} ∥ P_{n} ∥_{2}^{2} (n \geq k) .$ The functions $w_{k}^{1 / 2} P_{n}^{(k)}$ are mutually orthogonal by step 2.1 and have squared norms $λ_{n, k} ∥ P_{n} ∥_{2}^{2}$ . Applying finite Bessel inequalities to $w_{k}^{1 / 2} f^{(k)}$ and then increasing the finite index set yields $\sum_{n \geq k} λ_{n, k} ∣ c_{n} ∣^{2} ∥ P_{n} ∥_{2}^{2} \leq \int_{- 1}^{1} w_{k} ∣ f^{(k)} ∣^{2} \leq ∥ f^{(k)} ∥_{2}^{2} .$

4.1F1F2step 3.1algebra∎

Density [F2] and the best-approximation property of finite orthogonal projections imply $Π_{M} f \to f$ in $L^{2}$ : first approximate $f$ by a polynomial and then take $M$ at least its degree. Orthogonality [F1] and Pythagoras give $∥ f - Π_{N} f ∥_{2}^{2} = \sum_{n > N} ∣ c_{n} ∣^{2} ∥ P_{n} ∥_{2}^{2}$ . Since $λ_{n, k}$ increases for $n \geq k$ , step 3.1 bounds this tail by $λ_{N + 1, k}^{- 1} ∥ f^{(k)} ∥_{2}^{2}$ , which is the stated factorial estimate. When $k = 0$ , the same finite Pythagoras identity directly gives contraction.

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Analytic derivative bounds give exponential polynomial approximation on a triangle

Statement

Let $S$ be the closed reference triangle $\overline{T}$ , where $T = {x, y > 0, x + y < 1}$ , or the closed interval $[0, 1]$ , and let $f : S \to R$ be real-valued. Write $P_{p} (S)$ for restrictions of real polynomials of total degree at most $p$ on the triangle and degree at most $p$ on the interval. Say that $f$ is smooth up to $S$ when every partial derivative on the interior has a continuous extension to $S$ . Suppose that for some $M > 0$ and $A \geq 1$ ,

∥ D^{α} f ∥_{L^{\infty} (S)} \leq M A^{∣ α ∣} ∣ α ∣! for every multi-index α .

Use $∥ g ∥_{W^{1, \infty} (S)} : = ∥ g ∥_{L^{\infty} (S)} + \sum_{i = 1}^{\dim S} ∥ \partial_{i} g ∥_{L^{\infty} (S)}$ . Then constants $C, b > 0$ , depending only on $A$ and the fixed reference domain, satisfy

\inf_{q \in P_{p} (S)} ∥ f - q ∥_{W^{1, \infty} (S)} \leq C M e^{- b p} (p \geq 1) .

For the interval branch there is also an endpoint-preserving polynomial $J_{p} f \in P_{p} ([0, 1])$ whose derivative is the $L^{2}$ orthogonal projection of $f^{'}$ onto $P_{p - 1}$ . It satisfies $∥ (J_{p} f)^{'} ∥_{2} \leq ∥ f^{'} ∥_{2}$ and $∥ f - J_{p} f ∥_{H^{1}} \leq C M e^{- b p}$ after changing $C, b$ by constants depending only on $A$ .

Facts & Assumptions

Given: The fixed reference interval or triangle, the smoothness and factorial derivative bound in the statement, and $p \geq 1$ .

[F1]

Taylor's formula with Lagrange remainder holds along every segment contained in an open convex set. (Multivariable Taylor formula with a Lagrange remainder along a line segment)

[F2]

Cauchy–Schwarz bounds the integral of a derivative over an interval of length one by its $L^{2}$ norm. (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs)

[F3]

Multi-index order, factorial, monomial powers, and canonical partial derivatives have the conventions in the cited definition. ( $C^{k}$ maps and multi-index derivative notation in Euclidean space)

[F4]

The fundamental theorem of calculus gives $\int_{0}^{1} f^{'} (t) d t = f (1) - f (0)$ when $f$ is continuously differentiable on the interval, as it is here. (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ )

Proof

1.1givenalgebra

For $0 < τ < 1$ and integer $n \geq 1$ define $B_{n, τ} (t) = \sum_{j = ⌈ n τ ⌉}^{n} (\binom{n}{j}) t^{j} (1 - t)^{n - j}$ . Then $0 \leq B \leq 1$ , $\deg B \leq n$ , and $∣ B^{'} ∣ \leq n$ . If $t \leq τ - δ$ for fixed $δ > 0$ , Chernoff's bound follows directly from $1_{j \geq n τ} \leq e^{z (j - n τ)}$ for $z > 0$ : $B_{n, τ} (t) \leq e^{- z n τ} (1 - t + t e^{z})^{n} .$ For sufficiently small fixed $z$ , the exponent per $n$ is negative uniformly because its derivative at $z = 0$ is $t - τ \leq - δ$ . Applying the same argument to the complementary lower tail gives exponential smallness of $1 - B$ on $t \geq τ + δ$ . Differentiation telescopes to one binomial atom times $n$ ; away from the threshold its probability is bounded by the same tail estimate with gap $δ / 2$ for all sufficiently large $n$ , and increasing the constant covers the finitely many smaller $n$ . Thus $B$ and $1 - B$ switch from exponentially small to $1 + O (e^{- c n})$ across a strip of fixed width, and their derivatives outside that strip are $O (n e^{- c n})$ .

2.1step 1.1construct

Choose $0 < ρ < 1 / (32 A)$ and a finite coordinate-grid cover of $S$ by boxes of radius $ρ$ with centers $a_{i} \in S$ , $1 \leq i \leq m$ . In each coordinate form a left increasing binomial factor with threshold $a_{i, j} - 3 ρ / 2$ and a right decreasing factor with threshold $a_{i, j} + 3 ρ / 2$ ; omit the left factor when its threshold is at most zero and the right factor when its threshold is at least one. Their product $b_{i}$ has degree at most $4 n$ , lies in $[0, 1]$ , equals $1 + O (e^{- c n})$ on the inner box of radius $ρ$ and its first derivatives there are $O (n e^{- c n})$ ; outside the outer box of radius $2 ρ$ its value and first derivatives are $O (n e^{- c n})$ . The omitted boundary factors require no suppression on a side where the outer box does not meet $S$ .

3.1F1F3step 2.1givenalgebra

Let $T_{i, s}$ be the total-degree- $s$ Taylor polynomial at $a_{i}$ . When $a_{i}$ is on the boundary, take limits of interior centers; all derivatives extend continuously to $S$ . On the intersection of the outer box and $S$ , convexity keeps the Taylor segment inside the triangle, so [F1] and the multinomial formula give $∣ f - T_{i, s} ∣ + ∣ \nabla (f - T_{i, s}) ∣ \leq C M (s + 1) (4 A ρ)^{s} \leq C M 2^{- s}$ after enlarging $C$ for small $s$ . On all of $S$ , the coefficient formula for $T_{i, s}$ gives $∣ T_{i, s} ∣ + ∣ \nabla T_{i, s} ∣ \leq C M (s + 1)^{2} D^{s}$ for a fixed $D > 1$ depending only on $A$ .

3.2step 2.1algebra

Set $w_{i} = b_{i} \prod_{j < i} (1 - b_{j})$ and $w_{0} = \prod_{j \leq m} (1 - b_{j})$ . Then $\sum_{i = 0}^{m} w_{i} = 1$ and $0 \leq w_{i} \leq 1$ . At every point of $S$ some inner box applies, so one factor $1 - b_{i}$ , together with its first derivatives, is exponentially small there; the product rule and $∣ b_{j}^{'} ∣ \leq C n$ give $∥ w_{0} ∥_{W^{1, \infty} (S)} \leq C n e^{- c n}$ . Outside the outer box for a fixed $i$ , the same product rule gives $∣ w_{i} ∣ + ∣ \nabla w_{i} ∣ \leq C n e^{- c n}$ .

4.1step 3.2step 3.1algebraconstruct

Put $q = \sum_{i = 1}^{m} w_{i} T_{i, s}$ . Since $f - q = w_{0} f + \sum_{i} w_{i} (f - T_{i, s})$ , the near-box and far-box bounds of steps 3.1 and 3.2 give $∥ f - q ∥_{W^{1, \infty} (S)} \leq C M {n 2^{- s} + n (s + 1)^{2} D^{s} e^{- c n}} .$ Choose fixed $ε > 0$ with $ε \leq 1$ and $ε \log D < c / 2$ , and let $s = ⌊ ε n ⌋$ . Then the right side is at most $C M e^{- b_{0} n}$ . Each $w_{i}$ has degree at most $4 m n$ , so $q$ has total degree at most $4 m n + s \leq (4 m + 1) n$ . Choose $n = ⌊ p / (4 m + 1) ⌋$ ; for the finitely many $p$ with $n = 0$ , the zero polynomial and an enlarged constant handle the claim. This proves the exponential estimate for every $p \geq 1$ .

5.1F2F4step 4.1givenalgebra∎

On the interval let $P$ be the $L^{2}$ orthogonal projection of $f^{'}$ onto $P_{p - 1}$ and put $J_{p} f (t) = f (0) + \int_{0}^{t} P (s) d s$ . The positive-definite finite polynomial Gram matrix gives a unique projection; Pythagoras gives $∥ P ∥_{2} \leq ∥ f^{'} ∥_{2}$ and $∥ f^{'} - P ∥_{2} \leq ∥ f^{'} - r ∥_{2}$ for every $r \in P_{p - 1}$ . Orthogonality to constants gives $\int_{0}^{1} P = \int_{0}^{1} f^{'}$ , and [F4] evaluates the latter as $f (1) - f (0)$ . Thus $J_{p} f (1) = f (0) + \int_{0}^{1} P = f (1)$ , while $J_{p} f (0) = f (0)$ by construction. The derivatives of $f^{'}$ satisfy a factorial bound with fixed constants $M^{'} = M A$ and $A^{'} = 2 A$ , since $(j + 1)! \leq 2^{j} j!$ for $j \geq 1$ . For $p \geq 2$ , apply step 4.1 to $f^{'}$ with degree $p - 1$ ; best approximation of the $L^{2}$ projection then gives $∥ f^{'} - P ∥_{2} \leq C M e^{- b (p - 1)}$ , which is at most $C^{'} M e^{- b^{'} p}$ after changing constants. For $p = 1$ , the derivative contraction and the bound $∥ f^{'} ∥_{2} \leq M A$ give the same asserted estimate after enlarging $C^{'}$ . Finally $f - J_{p} f = \int_{0}^{t} (f^{'} - P)$ and [F2] bounds its $L^{2}$ norm by $∥ f^{'} - P ∥_{2}$ . The derivative and value bounds prove the stated interval estimate.

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Weighted analytic polygon functions have corner decay and scaled analytic bounds

Statement

Assume Countable Choice. Let $Ω \subset R^{2}$ be a bounded open polygonal domain with finitely many straight sides and corner set $C = {c_{1}, \dots, c_{m}}$ , with interior angle at each corner strictly between $0$ and $2 π$ . Let $0 \leq β_{i} < 1$ and write $Φ_{γ} (x) = \prod_{i = 1}^{m} ∣ x - c_{i} ∣^{γ_{i}}$ for a vector $γ \in [0, \infty)^{m}$ . For integer $k \geq 2$ , define

∥ v ∥_{H_{β}^{k, 2} (Ω)}^{2} = ∥ v ∥_{H^{1} (Ω)}^{2} + \sum_{j = 2}^{k} \sum_{∣ α ∣ = j} ∥ Φ_{β + j - 2} D^{α} v ∥_{L^{2} (Ω)}^{2},

where $(β + j - 2)_{i} = β_{i} + j - 2$ and a zero exponent contributes the factor one. Let $u \in H_{β}^{k, 2} (Ω)$ for every integer $k \geq 2$ , with real-valued weak derivatives, and suppose constants $C_{u}, d_{u} > 0$ satisfy

{∥ Φ_{β + j - 2} \sum_{∣ α ∣ = j} ∣ D^{α} u ∣ ∥}_{L^{2} (Ω)} \leq C_{u} d_{u}^{j - 2} (j - 2)! (j \geq 2) .

For each fixed $0 < a < \min_{i} (1 - β_{i})$ there are constants $C, A, R > 0$ such that the weak class of $u$ has a unique representative $\tilde{u} \in C (\overline{Ω})$ , smooth up to every open side away from corners, and for $0 < r = ∣ x - c_{i} ∣ < R$ ,

∣ \tilde{u} (x) - \tilde{u} (c_{i}) ∣ \leq C r^{a}, ∣ D^{α} \tilde{u} (x) ∣ \leq C A^{∣ α ∣} ∣ α ∣! r^{a - ∣ α ∣} (∣ α ∣ \geq 1) .

On every compact subset of $\overline{Ω} ∖ C$ , the derivatives satisfy ordinary factorial bounds $∣ D^{α} u ∣ \leq C_{0} A_{0}^{∣ α ∣} ∣ α ∣!$ . On every closed nondegenerate shape-regular affine triangle $K \subset \overline{Ω}$ with $K^{\circ} \subset Ω$ and $K \cap C = \emptyset$ , whose ratio $h_{K} / dist (K, C)$ is bounded above and below by fixed positive constants, where $dist (K, C) = \inf_{x \in K, c \in C} ∣ x - c ∣$ , set $r_{K} = \min (R, dist (K, C))$ . After subtracting a suitable constant from the pullback $\hat{u}$ to the reference triangle of Matching affine simplicial meshes and shape regularity, $∥ D^{α} \hat{u} ∥_{\infty} \leq C A_{1}^{∣ α ∣} ∣ α ∣! r_{K}^{a}$ for every multi-index $α$ , with constants uniform over this family. If a closed nondegenerate shape-regular affine triangle $K \subset \overline{Ω}$ with $K^{\circ} \subset Ω$ , diameter $h < R$ , has a corner $c_{i}$ as a vertex, its vertex affine interpolant $a_{K}$ satisfies

∣ u - a_{K} ∣_{H^{1} (K)} \leq C h^{a}, ∥ u - a_{K} ∥_{L^{2} (K)} \leq C h^{a + 1} .

Facts & Assumptions

Given: Countable Choice; the bounded polygon and its finite corner set; the weights $β_{i}$ ; the ordinary Sobolev class conventions of Integer-order Sobolev spaces and their norms; the stated weighted factorial bounds; and a fixed exponent $a$ as above.

[F1]

Fubini applies to integrable functions on the finite rectangle patches used below. (Fubini's theorem for L^1 functions on a sigma-finite product)

[F2]

Cauchy–Schwarz controls rectangle integrals, and the affine shape-regular bounds control pullbacks to triangles. (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs, Shape regularity controls affine simplex geometry)

[F3]

A nonnegative smooth compactly supported bump equal to one on a smaller ball has finite positive integral; after normalization its dilates form an $L^{1}$ approximate identity. Convolution with each dilate is smooth and converges in $L^{2}$ together with weak derivatives through order two. (A smooth bump between concentric Euclidean balls, The mollifier family generated by a unit-mass smooth bump, An $L^{1}$ approximate identity on $R^{n}$ , A unit-mass smooth bump generates an $L^{1}$ approximate identity, Every $L^{1}$ approximate identity converges to the identity in $L^{p}$ for $1 \leq p < \infty$ , Convolution with a mollifier is smooth, and derivatives pass under the integral sign)

[A1]

Countable Choice is exactly the choice assumption in the mollifier measure interfaces and the selection of the compatible representatives of the countable family of weak derivatives. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F4]

A nonempty open rectangle has positive Lebesgue measure; hence continuous representatives that agree almost everywhere agree on the interior and then on the closure. (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included)

Proof

1.1A1F1F2F3givenconstruct

On the unit square, for smooth $v$ the one-dimensional identity $∣ g (t) ∣ \leq \int_{0}^{1} ∣ g (s) ∣ d s + \int_{0}^{1} ∣ g^{'} (s) ∣ d s$ applied first in $x$ and then in $y$ gives $∥ v ∥_{\infty} \leq ∥ v ∥_{L^{1}} + ∥ v_{x} ∥_{L^{1}} + ∥ v_{y} ∥_{L^{1}} + ∥ v_{x y} ∥_{L^{1}} \leq \sum_{i, j = 0}^{1} ∥ \partial_{x}^{i} \partial_{y}^{j} v ∥_{L^{2}}$ by [F2]. For $v \in H^{2} ((0, 1)^{2})$ extend across $x = 0$ by $E v (x, y) = 3 v (- x, y) - 2 v (- 2 x, y)$ for small $x < 0$ , and across $x = 1$ by $E v (x, y) = 3 v (2 - x, y) - 2 v (3 - 2 x, y)$ for small $x > 1$ ; use the same two formulas in $y$ . The coefficients satisfy $3 - 2 = 1$ for values and $- 3 + 4 = 1$ for first normal derivatives. For a smooth test $ϕ$ across $x = 0$ , changing variables in the zeroth-order pairing gives $ψ_{0} (t) = ϕ (t) - 3 ϕ (- t) + 2 ϕ (- t / 2)$ , so $ψ_{0} (0) = 0$ . For the first normal derivative, the corresponding test combination is $ψ_{1} (t) = ϕ (t) + 3 ϕ (- t) - 4 ϕ (- t / 2)$ , also with $ψ_{1} (0) = 0$ . These formulas follow by rewriting the reflected integrals over $t > 0$ and applying the chain rule. Multiply each combined test by a cutoff that is zero on $(0, ε)$ and one on $(2 ε, \infty)$ before using the weak derivative identity on the original open square. If $∣ χ_{ε}' ∣ \leq C / ε$ , then $∣ ψ_{j} χ_{ε}' ∣ \leq C$ on the strip because $ψ_{j} (t) = O (t)$ ; for any relevant $g \in L^{2}$ , Cauchy--Schwarz bounds the cutoff error by $C \sqrt{2 ε} ∥ g ∥_{L^{2} (0, 2 ε)} \leq C \sqrt{2 ε} ∥ g ∥_{L^{2} (0, 1)} \to 0$ . Thus the reflected piecewise formulas are the weak derivatives through order two, including mixed derivatives. Changes of variables bound each derivative norm by a fixed multiple of $∥ v ∥_{H^{2}}$ . Reflect successively across the four sides, cut off in the enlarged square, and mollify. For every weak derivative through order two, inserting a translated compactly supported kernel into its weak derivative identity gives $D^{α} (v * φ_{ε}) = (D^{α} v) * φ_{ε}$ on the original square: the sign from integration by parts cancels the sign from differentiating the kernel in its integration variable. Thus [F3] gives smooth approximants converging in $H^{2}$ . The smooth inequality makes them uniformly Cauchy, yielding a continuous representative and the same bound for every $H^{2}$ class. Scaling gives $∥ v ∥_{L^{\infty} (Q)} \leq C (r^{- 1} ∥ v ∥_{2} + ∥ D v ∥_{2} + r ∥ D^{2} v ∥_{2})$ on a rectangle of side comparable to $r$ ; a parallelogram with fixed aspect ratio has the same estimate after an affine change of variables.

2.1A1F1F2F3F4step 1.1givenalgebra

Choose $R > 0$ so distinct corner balls of radius $4 R$ are disjoint and each meets only the two sides at its corner. For a point $x$ with $r = ∣ x - c_{i} ∣ < R$ , choose an interior rectangle, or near one straight side an inward rectangle, of diameter comparable to $r$ and lying in $r / 2 < ∣ z - c_{i} ∣ < 2 r$ . A finite angular cover gives aspect ratios and widths bounded in terms of the fixed corner angles. For each individual weak derivative $g_{α} = D^{α} u$ with $∣ α ∣ = j \geq 2$ , the weighted hypotheses imply $g_{α} \in H^{2}$ on this rectangle. Apply the fixed second-order rectangle extension and estimate of step 1.1 to $g_{α}$ itself. In particular, the coefficients $3, - 2$ are not used to extend $u$ in arbitrary Sobolev orders and then differentiated across a reflecting side. The extension of each $g_{α}$ is needed only through order two, so its bound is fixed, independently of $j$ . On the annulus the other corner distances are bounded above and below, so at derivative order $s \geq 2$ , $Φ_{β + s - 2} \geq C^{- s} r^{β_{i} + s - 2}$ . The weighted hypotheses at orders $j, j + 1, j + 2$ therefore bound $∥ g_{α} ∥_{2}$ , $∥ D g_{α} ∥_{2}$ , and $∥ D^{2} g_{α} ∥_{2}$ respectively by $C A^{j} j! r^{2 - β_{i} - j}$ , $C A^{j} j! r^{1 - β_{i} - j}$ , and $C A^{j} j! r^{- β_{i} - j}$ . Fixed affine changes of coordinates contribute at most another fixed base to the power $j$ ; the boundedly many extra derivatives and their factorial factors are absorbed into $A^{j}$ . Multiplying these three norms by $r^{- 1}, 1, r$ as in step 1.1 gives $∣ D^{α} u (x) ∣ \leq C A^{j} j! r^{1 - β_{i} - j} \leq C A^{j} j! r^{a - j}$ , uniformly up to an open side. On compact patches away from corners the same argument gives ordinary factorial derivative bounds. There apply step 1.1 also to $g = u$ and to each first derivative: their lower-order $L^{2}$ terms come from $u \in H^{1}$ , while derivative orders two and three are bounded by the weighted hypotheses since the weights are bounded below on those patches. Thus $u$ and its gradient are bounded there, in particular on the fixed radius- $R$ arcs. These separately constructed continuous representatives of the weak derivatives agree on overlaps by [F4]. They are compatible as derivatives: on a smaller interior patch mollify $u$ itself, so each derivative converges locally in $H^{2}$ and therefore uniformly by step 1.1 to the representative of that weak derivative; the fundamental theorem along interior line segments passes to these uniform limits. All derivative identities then extend to an open side by continuity. Thus $u$ is smooth up to sides away from corners without an arbitrary-order reflected extension.

3.1step 2.1algebra

On the fixed arc of radius $R$ about $c_{i}$ , step 2.1 bounds $\nabla u$ . Along every ray, the Hessian bound in step 2.1 gives $∣ \nabla u (r, θ) ∣ \leq C + C \int_{r}^{R} t^{- 1 - β_{i}} d t$ . If $β_{i} > 0$ this is at most $C r^{- β_{i}}$ ; if $β_{i} = 0$ it is at most $C (1 + ∣ \log r ∣)$ . Since $a < 1 - β_{i}$ , both are bounded by $C r^{a - 1}$ for $r < R$ after decreasing $R$ if needed. Integrating radially shows that $u (r, θ)$ has a limit as $r ↓ 0$ and differs from it by at most $C r^{a}$ . Variation along the circular arc of radius $r$ is at most $C r \cdot r^{a - 1} = C r^{a}$ , so the radial limits agree for all angles; call the common value $u (c_{i})$ . This also proves continuity at the corner. The first-derivative estimate gives the claimed bound for $∣ α ∣ = 1$ , while step 2.1 gives it for higher orders because $r^{1 - β_{i} - j} \leq r^{a - j}$ .

4.1F2step 2.1step 3.1algebra

For a regular triangle $K$ in the stated family, when its corner distance is sufficiently small compared with $R$ and the fixed upper bound for $h_{K} / dist (K, C)$ , all its points lie in a single corner neighborhood; choose the constant $u (c_{i})$ there; the pointwise derivative estimate and $h_{K} / dist (K, C) ≍ 1$ give $h_{K}^{∣ α ∣} ∥ D^{α} u ∥_{\infty} \leq C A_{1}^{∣ α ∣} ∣ α ∣! r_{K}^{a}$ after affine pullback and the chain rule. For the remaining triangles, their distance from all corners is bounded below by a fixed positive constant; choose the value at one point of $K$ as the constant; the compact-patch derivative bounds and bounded diameter give the same estimate after increasing $C, A_{1}$ . The affine geometry factors are controlled by [F2].

5.1F2step 3.1givenalgebra∎

Let $K$ have vertex $c_{i}$ and diameter $h < R$ . On dyadic annuli $2^{- j - 1} h \leq ∣ x - c_{i} ∣ \leq 2^{- j} h$ , step 3.1 bounds $∣ \nabla u ∣^{2}$ by $C (2^{- j - 1} h)^{2 a - 2}$ , while the area of each annulus is at most $C (2^{- j} h)^{2}$ . Summing the resulting geometric series gives $∥ \nabla u ∥_{L^{2} (K)}^{2} \leq C h^{2 a}$ , hence $∥ \nabla u ∥_{2} \leq C h^{a}$ . Every vertex value differs from $u (c_{i})$ by at most $C h^{a}$ ; the barycentric gradients are bounded by $C / h$ and $∣ K ∣ \leq C h^{2}$ by shape regularity, so the vertex affine interpolant has gradient norm at most $C h^{a}$ . The pointwise value estimate gives both $L^{2}$ errors at most $C h^{a + 1}$ . The triangle inequality proves the stated seminorm and full $L^{2}$ bounds.

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Legendre-Gauss-Lobatto nodes, weights, and tensor quadrature

Definition

Fix an integer $p \geq 1$ . Let $ξ_{0} = - 1$ , $ξ_{p} = 1$ , and let $ξ_{1} < \dots < ξ_{p - 1}$ be the roots of $P_{p}^{'}$ in $(- 1, 1)$ ; for $p = 1$ this interior list is empty. By Legendre-Gauss-Lobatto nodes are distinct and interior critical points are simple these $p + 1$ nodes are distinct. Define $ℓ_{j} (x) = \prod_{0 \leq i \leq p, i \neq j} (x - ξ_{i}) / (ξ_{j} - ξ_{i})$ , set $w_{j} = \int_{- 1}^{1} ℓ_{j} (x) d x$ , and define the Legendre–Gauss–Lobatto rule by $Q_{p} (f) = \sum_{j = 0}^{p} w_{j} f (ξ_{j})$ .

On $[- 1, 1]^{d}$ its tensor rule is $Q_{p}^{\otimes d} (f) = \sum_{j_{1}, \dots, j_{d} = 0}^{p} (w_{j_{1}} \dots w_{j_{d}}) f (ξ_{j_{1}}, \dots, ξ_{j_{d}})$ . For an affine box map $F_{K} (ξ) = b_{K} + J_{K} ξ$ , define the transported rule by $Q_{K, p} (f) = ∣ \det J_{K} ∣ Q_{p}^{\otimes d} (f \circ F_{K})$ . The nodal basis on $K$ is the pullback of $\prod_{r = 1}^{d} ℓ_{j_{r}} (ξ_{r})$ . The local mass and stiffness forms are respectively $m_{K, p} (v, w) = Q_{K, p} (v w)$ and $a_{K, p} (v, w) = Q_{K, p} (\nabla v \cdot \nabla w)$ ; any sampled coefficient is evaluated at the same quadrature nodes. Global forms are assembled by summing the cell forms.

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Uniform-degree polygon corner spaces have cubic degree-of-freedom growth

Statement

For $k \geq 1$ , let a matching triangulation $T_{k}$ of a polygon have at most $C_{0} k$ triangles. Assign a uniform total polynomial degree $p_{k} \geq 1$ with $p_{k} \leq C_{1} k$ . The algebraic space of continuous piecewise $P_{p_{k}}$ functions has dimension

N_{k} \leq C_{2} k^{3},

where $C_{2}$ depends only on $C_{0}$ and $C_{1}$ . The same upper bound holds for any subspace imposing homogeneous boundary traces. This is an upper count only; it asserts no hp approximation rate.

Facts & Assumptions

Given: Integer $k \geq 1$ , a matching triangulation $T_{k}$ with $# T_{k} \leq C_{0} k$ , and uniform degree $1 \leq p_{k} \leq C_{1} k$ .

[F1]

Variable-degree conforming hp spaces on matching affine meshes: the continuous polynomial space is a linear subspace of the broken product of its cell polynomial spaces.

Proof

1.1F1givenalgebra

On a triangle, monomials $x^{i} y^{j}$ with $i, j \geq 0$ and $i + j \leq p_{k}$ form a basis of $P_{p_{k}}$ . Counting by total degree gives $\dim P_{p_{k}} = \sum_{r = 0}^{p_{k}} (r + 1) = (p_{k} + 1) (p_{k} + 2) / 2$ .

2.1F1step 1.1givenalgebra∎

The broken product over $T_{k}$ therefore has dimension $# T_{k} (p_{k} + 1) (p_{k} + 2) / 2$ . By [F1], trace matching and any zero-boundary conditions define a linear subspace, whose dimension cannot exceed the broken dimension. Using $# T_{k} \leq C_{0} k$ and $p_{k} \leq C_{1} k$ , and $k \geq 1$ , gives $N_{k} \leq C_{0} k (C_{1} k + 1) (C_{1} k + 2) / 2 \leq C_{2} k^{3}$ for an explicit finite $C_{2}$ depending only on $C_{0}, C_{1}$ .

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Matching variable-degree polynomial traces give an H1 function

Statement

Assume Countable Choice. On a finite matching affine simplicial mesh or a finite face-to-face affine tensor-box mesh, a piecewise polynomial scalar function whose two traces agree on every interior facet belongs to $H^{1} (Ω)$ , and its weak gradient equals its ordinary cellwise polynomial gradient almost everywhere. The simplex degrees may vary by cell; the box degrees may also vary.

Facts & Assumptions

Given: Countable Choice; a finite matching mesh $T$ of affine simplices or affine tensor boxes; a real-valued function $v$ whose restriction to each cell is a polynomial of its assigned finite degree; equality of the two traces on every interior facet; and a test function $φ \in C_{c}^{\infty} (Ω)$ .

[F1]

Variable-degree conforming hp spaces on matching affine meshes: the mesh has finitely many cells, each cell restriction is a polynomial, and the conforming condition is equality of the polynomial traces on every whole shared facet.

[F2]

Continuous matching piecewise polynomials belong to W1p: under Countable Choice, a finite matching affine simplicial mesh with a single finite polynomial degree and matching traces gives a member of $W^{1, 2}$ whose weak gradient is cellwise.

[F3]

Divergence for finite piecewise C1 presentations: under Countable Choice, cellwise Gauss–Green holds for a finite piecewise $C^{1}$ presentation, and the fluxes on a shared face cancel when the pieces are glued.

[F4]

A weak derivative is characterized by integration against compactly supported smooth tests, and $H^{1} = W^{1, 2}$ consists of $L^{2}$ classes with all first weak derivatives in $L^{2}$ . (Weak derivative of a locally integrable function, Integer-order Sobolev spaces and their norms)

Proof

1.1F1F2givenchoose

If the mesh is empty, its domain is empty by the mesh convention and the assertion is immediate. Otherwise, because the mesh and every local degree are finite, $v$ and each cellwise first derivative $g_{i} : = \partial_{i} (v ∣_{K})$ are bounded on each closed cell; a finite union then puts $v, g_{i}$ in $L^{2} (Ω)$ . If the mesh is simplicial, set $p_{*} = \max_{K \in T} p_{K}$ . Every restriction lies in $P_{p_{*}} (K)$ , and the trace-matching condition is unchanged. Applying [F2] at the single degree $p_{*}$ proves the claim on this branch.

1.2F1F3givenalgebra

Suppose the cells are affine tensor boxes. Fix a coordinate $i$ . On each cell, apply [F3] to the smooth field $v φ e_{i}$ ; its divergence is $v \partial_{i} φ + g_{i} φ$ . Sum the finitely many resulting identities. At an interior face the two outward normals are opposite, because the adjacent cell interiors lie on opposite sides of its supporting hyperplane; the common trace in [F1] therefore makes the two fluxes cancel. The boundary-face terms vanish since $φ$ has compact support in $Ω$ . Thus $\int_{Ω} v \partial_{i} φ = - \int_{Ω} g_{i} φ$ .

2.1F1F4step 1.1step 1.2algebra∎

By [F4], step 1.2 says that $g_{i}$ is the weak $i$ th derivative of $v$ . This holds for every coordinate $i$ , and step 1.1 supplies $v, g_{i} \in L^{2}$ ; hence $v \in H^{1} (Ω)$ with the stated cellwise weak gradient. Together with the simplex branch of step 1.1 this proves the assertion for both mesh types.

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Integrated Legendre projection gives an explicit p-rate in H1

Statement

Assume Countable Choice. Let $p \geq 1$ , $s \geq 0$ , $g \in H_{L}^{s}$ as defined in Legendre weighted spectral-Sobolev scale and projection, identified with its canonical $L^{2} (- 1, 1)$ class, and let $c \in R$ . Define $u (x) = c + \int_{- 1}^{x} g (t) d t$ and $I_{p} u = I_{p} (c, g)$ by Endpoint-preserving integrated Legendre projector. Then $u$ and $I_{p} u$ are absolutely continuous, $I_{p} u \in P_{p}$ , and $I_{p} u (\pm 1) = u (\pm 1)$ . Moreover, $∥ u - I_{p} u ∥_{H^{1} (- 1, 1)} \leq \sqrt{5} (1 + p (p + 1))^{- s / 2} ∥ g ∥_{H_{L}^{s}},$ where the displayed $H^{1}$ error norm is $(∥ u - I_{p} u ∥_{L^{2}}^{2} + ∥ g - Π_{p - 1} g ∥_{L^{2}}^{2})^{1 / 2}$ . This coefficient-scale assertion does not identify $H_{L}^{s}$ with an ordinary derivative-based Sobolev norm.

Facts & Assumptions

Given: Countable Choice, $p \geq 1$ , $s \geq 0$ , $g \in H_{L}^{s}$ , and $c \in R$ .

[F1]

Endpoint-preserving integrated Legendre projector: $I_{p} (c, g) = c + \int_{- 1}^{x} Π_{p - 1} g$ is the degree- $p$ endpoint-anchored operator, with $Π_{p - 1}$ the finite Legendre projection.

[F2]

Legendre polynomials have explicit norms and weighted derivative orthogonality: $\int_{- 1}^{1} P_{n} P_{m} = 2 δ_{n m} / (2 n + 1)$ and $P_{0} = 1$ .

[F3]

Legendre weighted spectral-Sobolev scale and projection: $H_{L}^{s}$ is a coefficient completion with weights $(1 + n (n + 1))^{s} 2 / (2 n + 1)$ , so truncation is the finite projection from [F1] and its coefficient norm dominates the $s = 0$ coefficient norm.

[F4]

Legendre weighted projection has its spectral-Sobolev rate: $∥ g - Π_{N} g ∥_{L_{L}^{2}} \leq (1 + (N + 1) (N + 2))^{- s / 2} ∥ g ∥_{H_{L}^{s}}$ .

[F5]

The indefinite integral of an $L^{1}$ function is absolutely continuous: the primitive of an $L^{1}$ function is absolutely continuous.

[F6]

The indefinite integral of an $L^{1}$ function is differentiable almost everywhere: under Countable Choice, the derivative of the primitive of an $L^{1}$ function equals its integrand almost everywhere.

[F7]

Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs: the $L^{2}$ inner product obeys Cauchy–Schwarz.

[F8]

Polynomials are uniformly dense in $C ([a, b], R)$ for every closed interval: polynomials are uniformly dense in $C ([- 1, 1])$ .

[F9]

Smooth compactly supported functions of an open set are dense in $L^{2}$ : $C_{c}^{\infty} ((- 1, 1))$ is dense in $L^{2} (- 1, 1)$ .

[F10]

Riesz-Fischer completeness of $L^{p}$ for $1 \leq p \leq \infty$ : under Countable Choice, $L^{2} (- 1, 1)$ is complete.

Proof

1.1F1F2F3F5F7F8F9F10givenalgebra

By [F2], finite Legendre sums span all polynomials. The density facts [F8] and [F9] imply that polynomials are dense in $L^{2} (- 1, 1)$ : first approximate an $L^{2}$ class by a compactly supported smooth function, then approximate that function uniformly on $[- 1, 1]$ by a polynomial. The coefficient norm at $s = 0$ is the ordinary $L^{2}$ norm on finite Legendre sums by [F2]–[F3], and [F10] makes $L^{2} (- 1, 1)$ complete. Hence the identity on finite sums extends to an isometry from $H_{L}^{0}$ onto $L^{2} (- 1, 1)$ . Since each spectral weight for $s \geq 0$ is at least its $s = 0$ weight, [F3] gives a continuous embedding $H_{L}^{s} \to H_{L}^{0}$ . Thus $g$ has the canonical $L^{2}$ class in the Statement, and Cauchy–Schwarz [F7] on the interval of length two gives $g \in L^{1}$ . Then [F5] makes $u$ absolutely continuous. The definition [F1] makes $I_{p} u$ the integral of a polynomial of degree at most $p - 1$ , hence a degree-at-most- $p$ polynomial. By the finite projection formula in [F1] and Legendre orthogonality [F2], $g - Π_{p - 1} g$ is orthogonal to constants, so $\int Π_{p - 1} g = \int g$ and the endpoint values agree. The primitive defining $I_{p} u$ is absolutely continuous by [F5].

2.1F1F3F4F6F7step 1.1algebra∎

Apply [F6] to the two primitives in step 1.1; their a.e. derivatives are $g$ and $Π_{p - 1} g$ , so $e = u - I_{p} u$ has derivative $e^{'} = g - Π_{p - 1} g$ . By [F3] the finite projection is the truncation in [F1]; the tail estimate [F4] with $N = p - 1$ gives $∥ e^{'} ∥_{L^{2}} \leq (1 + p (p + 1))^{- s / 2} ∥ g ∥_{H_{L}^{s}}$ . Since $e (- 1) = 0$ , $e (x) = \int_{- 1}^{x} e^{'} (t) d t$ ; Cauchy–Schwarz [F7] and integration over an interval of length two give $∥ e ∥_{L^{2}} \leq 2 ∥ e^{'} ∥_{L^{2}}$ . Therefore $∥ e ∥_{H^{1}}^{2} = ∥ e ∥_{L^{2}}^{2} + ∥ e^{'} ∥_{L^{2}}^{2} \leq 5 ∥ e^{'} ∥_{L^{2}}^{2}$ , proving the estimate.

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Degree-explicit polynomial trace inverse inequality on simplices

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )), as required by the affine-scaling supplier used below.

Let $K$ be a nondegenerate $d$ -simplex, $d \geq 2$ , let $F$ be one of its facets, and let $p \geq 0$ . For every $v \in P_{p} (K)$ ,

∥ v ∥_{L^{2} (F)}^{2} \leq \frac{(p + 1) (p + d) ∣ F ∣}{d ∣ K ∣} ∥ v ∥_{L^{2} (K)}^{2} .

If $K$ belongs to a shape-regular affine simplex family with $h_{K} = diam (K)$ and shape bound $σ$ , then

∥ v ∥_{L^{2} (F)}^{2} \leq C_{d, σ} \frac{(p + 1) (p + d)}{h_{K}} ∥ v ∥_{L^{2} (K)}^{2} .

For $p \geq 1$ , the normal derivative of $v$ has degree at most $p - 1$ on $K$ and satisfies the same inequalities with $(p + 1) (p + d)$ replaced by $p (p + d - 1)$ . In particular, polynomial normal-flux trace penalties have the explicit scale $p (p + d - 1) / h_{K}$ ; this is an inverse trace bound, not a p-approximation or a DG coercivity theorem.

Facts & Assumptions

Given: Countable Choice, a nondegenerate simplex $K$ , a face $F$ , an integer $p \geq 0$ , and, for the scaled inequality, a shape-regular family.

[F1]

Weighted Jacobi endpoint kernel on the reference interval: every degree- $m$ polynomial $q$ on $[0, 1]$ satisfies $∣ q (0) ∣^{2} \leq (m + 1) (m + d) \int_{0}^{1} (1 - t)^{d - 1} ∣ q (t) ∣^{2} d t$ .

[F2]

The altitude from the opposite vertex $a$ to $F$ is $H_{F} = d ∣ K ∣ / ∣ F ∣$ , and the map $x = (1 - t) y + t a$ has volume Jacobian $H_{F} (1 - t)^{d - 1}$ for $y \in F$ , $0 < t < 1$ .

[F3]

Shape regularity controls affine simplex geometry: under shape bound $σ$ , $∣ F ∣ / ∣ K ∣ \leq C_{d, σ} / h_{K}$ . The supplier assumes Countable Choice.

[F4]

Fubini's theorem for L^1 functions on a sigma-finite product: the simplex-pyramid parameterization in [F2] permits integrating the bounded polynomial integrand first along $t$ and then over $F$ .

[A1]

The mesh and polynomial-degree conventions for affine simplices are those of Variable-degree conforming hp spaces on matching affine meshes; in particular the unit normal to a flat face is constant.

Proof

1.1F1F2F4givenalgebra

Fix $y \in F$ and let $q_{y} (t) = v ((1 - t) y + t a)$ . This is a polynomial of degree at most $p$ in $t$ , and $q_{y} (0) = v (y)$ . Apply [F1] with $m = p$ , then integrate the resulting inequality over $F$ . In the map $(y, t) \mapsto (1 - t) y + t a$ , each tangent direction in $F$ is scaled by $1 - t$ , while the remaining component is the altitude $H_{F}$ ; hence the absolute volume Jacobian is $H_{F} (1 - t)^{d - 1}$ , as stated in [F2]. By Fubini [F4], $∥ v ∥_{L^{2} (F)}^{2} \leq (p + 1) (p + d) \int_{F} \int_{0}^{1} (1 - t)^{d - 1} ∣ v ((1 - t) y + t a) ∣^{2} d t d S_{y} = \frac{(p + 1) (p + d) ∣ F ∣}{d ∣ K ∣} ∥ v ∥_{L^{2} (K)}^{2}$ . This proves the exact reference-geometric inequality.

2.1A1F1F2step 1.1

The cellwise normal derivative $\partial_{n} v = n_{F} \cdot \nabla v$ is a polynomial of degree at most $p - 1$ . If $p = 0$ it is zero. If $p \geq 1$ , repeat step 1.1 with degree $m = p - 1$ ; the endpoint factor becomes $p (p + d - 1)$ .

3.1F3step 1.1step 2.1algebra∎

In a shape-regular family, apply [F3] to the factor $∣ F ∣ / ∣ K ∣$ in step 1.1 and in the same argument for $\partial_{n} v$ . This gives both $h_{K}^{- 1}$ estimates. The only declared choice assumption is the Countable Choice required by [F3]; [F4] adds no choice hypothesis, and the linewise kernel uses finite-dimensional algebra.

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Constructive ordinary Sobolev polynomial approximation on triangles

Statement

Assume the Axiom of Choice. Let $T = {(x, y) : x > 0, y > 0, x + y < 1}$ , let $r \geq 5$ be an integer, let $p \geq 2 r - 4$ be an integer, and put $n = ⌊ p / 2 ⌋$ and

η_{n, r} = {(\frac{(n - r + 3)!}{(n + r - 5)!})}^{1 / 2} .

There is a bounded linear map $A_{p, r} : H^{r} (T; R) \to P_{p} (T; R)$ , where $P_{p} (T; R)$ is the restriction of real polynomials in $(x, y)$ of total degree at most $p$ . It is explicitly defined in the proof from the reference-simplex extension, a unique moment polynomial, and finite Legendre projections, and satisfies

∥ u - A_{p, r} u ∥_{H^{2} (T)} \leq C_{r} η_{n, r} ∣ u ∣_{H^{r} (T)} .

Here $C_{r}$ is independent of $p$ and is the product of a fixed reference extension bound, the reference moment-projection bound, and an absolute tensor-projection constant; no useful growth bound in $r$ is asserted. For a shape-regular affine triangle $K$ of diameter $h_{K}$ , affine transport gives

{(\sum_{j = 0}^{2} h_{K}^{2 j - 2} ∣ u - A_{K, p, r} u ∣_{H^{j} (K)}^{2})}^{1 / 2} \leq C_{r, γ} η_{n, r} h_{K}^{r - 1} ∣ u ∣_{H^{r} (K)} .

If $p \geq 4 (r - 2)$ , then $η_{n, r} \leq (4 / p)^{r - 4}$ . This is a conservative ordinary Sobolev rate; it is neither an optimal $p^{- (r - 1)}$ estimate nor an analytic growth estimate for $C_{r}$ .

Facts & Assumptions

Given: Full Choice; the reference triangle $T$ ; integers $r \geq 5$ , $p \geq 2 r - 4$ ; and $u \in H^{r} (T; R)$ with weak derivatives and Sobolev norms as in Integer-order Sobolev spaces and their norms.

[A1]

Full Choice is used by the reference-simplex extension and by the one-dimensional Sobolev representative supplier. (The Axiom of Choice)

[F1]

The ordinary Legendre projection obeys the factorial $L^{2}$ derivative-tail estimate for every integer derivative order. The endpoint-anchored operator $S_{n}$ used below is defined locally from this projection. (An ordinary derivative controls the Legendre tail with a factorial ratio)

[F2]

The reference simplex has a bounded linear extension $E_{r} : H^{r} (T) \to H^{r} (R^{2})$ ; write $M_{r} = ∥ E_{r} ∥$ . (A reference simplex is an integer-order Sobolev extension domain)

[F3]

The unique polynomial $B_{r - 1} u \in P_{r - 1}$ whose derivative moments through order $r - 1$ match those of $u$ satisfies $∥ u - B_{r - 1} u ∥_{H^{r} (T)} \leq B_{r} ∣ u ∣_{H^{r} (T)}$ . (Bramble–Hilbert approximation on a reference simplex)

[F4]

Smooth functions on $R^{2}$ are dense in $H^{r} (R^{2})$ , and one-dimensional weak derivatives through order $s - 1$ have absolutely continuous representatives on the interval. (Compactly supported smooth functions are dense in W^{k,p}(R^n), One-dimensional $W^{1, p}$ functions have unique absolutely continuous representatives)

[F5]

Fubini and Cauchy–Schwarz justify the tensor-section norms and primitive bounds. (Fubini's theorem for L^1 functions on a sigma-finite product, Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs)

[F6]

Affine pullback derivatives and Sobolev seminorms scale with the singular values, determinant, and shape-regularity bound. (Shape regularity controls affine simplex geometry)

[F7]

The finite-dimensional tensor polynomial space $Q_{n}$ is closed in $H^{2} (I^{2})$ . (A finite-dimensional normed subspace is closed)

Proof

1.1A1F1F4F5givenconstructalgebra

On $I = (- 1, 1)$ define $S_{n} f (x) = f (- 1) + (x + 1) f^{'} (- 1) + \int_{- 1}^{x} (x - t) Π_{n - 2} f^{''} (t) d t,$ where $Π_{n - 2}$ is the ordinary Legendre $L^{2}$ projection. Here $p \geq 2 r - 4$ implies $n \geq r - 2 \geq 3$ . For $f \in H^{2} (I)$ , the endpoint values are those of its absolutely continuous representatives from [F4]. The polynomial $S_{n} f$ has degree at most $n$ and matches $f, f^{'}$ at $- 1$ . Its second derivative is an $L^{2}$ contraction; integrating twice and using the one-dimensional endpoint estimate gives $∥ S_{n} f ∥_{H^{2} (I)} \leq C ∥ f ∥_{H^{2} (I)}$ with absolute $C$ . For every integer $s \geq 3$ with $n \geq s - 1$ , the error $e = f - S_{n} f$ has $e (- 1) = e^{'} (- 1) = 0$ and $e^{''} = f^{''} - Π_{n - 2} f^{''}$ . Apply the Legendre tail estimate to $f^{''}$ with derivative order $s - 2$ and projection degree $n - 2$ ; the factorial ratio is $ρ_{n, s} = {(\frac{(n - s + 1)!}{(n + s - 3)!})}^{1 / 2} .$ Two integrations of $e^{''}$ , with Cauchy–Schwarz [F5], give $∥ e ∥_{H^{2} (I)} \leq C ρ_{n, s} ∥ f^{(s)} ∥_{2}$ . The same inequality holds for each derivative of order at most two.

2.1F1F4F5F7step 1.1algebra

Put $Q_{n} = span {x^{i} y^{j} : 0 \leq i, j \leq n}$ , apply $S_{n}$ in each coordinate on $I^{2}$ , and write $S_{n}^{x}, S_{n}^{y}$ . For smooth $U$ , $U - S_{n}^{x} S_{n}^{y} U = (1 - S_{n}^{x}) U + S_{n}^{x} (1 - S_{n}^{y}) U .$ Let $a + b \leq 2$ . For the first term choose $s_{b} = r - \max (1, b)$ , so $s_{b} \in {r - 1, r - 2}$ , $s_{b} \geq 3$ , $n \geq s_{b} - 1$ , and $s_{b} + b \leq r$ . Step 1.1 on the $x$ sections of $D_{y}^{b} U$ bounds $∥ D_{x}^{a} D_{y}^{b} (1 - S_{n}^{x}) U ∥_{2}$ by $C ρ_{n, s_{b}} ∥ D_{x}^{s_{b}} D_{y}^{b} U ∥_{2}$ . For the second term, the uniform $H_{x}^{2}$ bound in step 1.1 bounds its $D_{x}^{a} D_{y}^{b}$ norm by $C \sum_{j = 0}^{2} ∥ D_{x}^{j} D_{y}^{b} (1 - S_{n}^{y}) U ∥_{2}$ . For each such $j$ , choose $t_{j} = r - \max (1, j) \in {r - 1, r - 2}$ ; then $t_{j} \geq 3$ , $n \geq t_{j} - 1$ , and $j + t_{j} \leq r$ . Apply the $H_{y}^{2}$ error estimate of step 1.1 to $D_{x}^{j} U$ ; it bounds every $y$ derivative of order $b \leq 2$ in this sum by $C ρ_{n, t_{j}} ∥ D_{x}^{j} D_{y}^{t_{j}} U ∥_{2}$ . Because $ρ_{n, r - 1} \leq ρ_{n, r - 2} = η_{n, r}$ , all terms are bounded by $C η_{n, r} ∥ U ∥_{H^{r} (I^{2})}$ . Fubini [F5] and addition over $a + b \leq 2$ give $∥ U - S_{n}^{x} S_{n}^{y} U ∥_{H^{2} (I^{2})} \leq C η_{n, r} ∥ U ∥_{H^{r} (I^{2})} .$ For the inputs used below, namely restrictions of $H^{r} (R^{2})$ extensions, choose smooth whole-space approximants by [F4]. The estimate on their differences makes $S_{n}^{x} S_{n}^{y} U$ a well-defined $H^{2}$ limit. Every smooth approximant has image in $Q_{n}$ , so closedness [F7] gives $S_{n}^{x} S_{n}^{y} U \in Q_{n}$ and the same bound for the limit.

3.1A1F2F3step 2.1constructalgebra

Let $b = B_{r - 1} u$ and $w = u - b$ . By [F3], $∥ w ∥_{H^{r} (T)} \leq B_{r} ∣ u ∣_{H^{r} (T)}$ . Set $U = (E_{r} w) ∣_{I^{2}}$ and define $A_{p, r} u = b + (S_{n}^{x} S_{n}^{y} U) ∣_{T} .$ This map is linear. The first summand has degree at most $r - 1 \leq p$ ; the second lies in $Q_{n} \subseteq P_{2 n} \subseteq P_{p}$ . Since $E_{r} w = w$ on $T$ , step 2.1 and [F2]–[F3] yield $∥ u - A_{p, r} u ∥_{H^{2} (T)} \leq C M_{r} B_{r} η_{n, r} ∣ u ∣_{H^{r} (T)} .$ This identifies the constant as the absolute tensor constant times the fixed extension and moment bounds; no growth in $r$ is asserted.

4.1F6step 3.1givenalgebra∎

Transport step 3.1 to a shape-regular affine triangle $K$ . For $e = u - A_{K, p, r} u$ and $\hat{e} = e \circ F_{K}$ , two-dimensional affine pullback gives $∣ \hat{e} ∣_{H^{j} (T)} ≍_{j, γ} h_{K}^{j - 1} ∣ e ∣_{H^{j} (K)}$ for $j = 0, 1, 2$ , and $∣ \hat{u} ∣_{H^{r} (T)} \leq C_{r, γ} h_{K}^{r - 1} ∣ u ∣_{H^{r} (K)}$ by [F6]. This proves the displayed full scaled estimate. Finally, $η_{n, r}^{- 2}$ is a product of $2 (r - 4)$ consecutive integers beginning with $n - r + 4$ . If $p \geq 4 (r - 2)$ and $n = ⌊ p / 2 ⌋$ , then $n - r + 4 \geq p / 4$ ; taking square roots yields $η_{n, r} \leq (4 / p)^{r - 4}$ .

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Analytic data converge exponentially under fixed-mesh p refinement

Statement

Assume Countable Choice. Let $Ω$ be partitioned by a fixed finite matching affine tensor-box mesh, and let $u \in C^{1} (\overline{Ω})$ be smooth on each box. For each cell $K$ , let ${\hat{u}}_{K}$ be its pullback to $[- 1, 1]^{d}$ . Assume that for fixed constants $M > 0$ and $R > 1$ ,

\sup_{\hat{x} \in [- 1, 1]^{d}} ∣ \partial^{ν} {\hat{u}}_{K} (\hat{x}) ∣ \leq M ∣ ν ∣! R^{- ∣ ν ∣}

for every multi-index $ν \in N_{0}^{d}$ and every cell $K$ . Choose any $q$ with $R^{- 1} < q < 1$ . For each common degree $p \geq 1$ , tensorize the endpoint-preserving integrated Legendre projector of Endpoint-preserving integrated Legendre projector on each reference cell and transport it to $K$ , obtaining $u_{p}$ . Then the cell traces of $u_{p}$ agree across every shared face and

∥ u - u_{p} ∥_{H^{1} (Ω)} \leq C_{T, d, R, q} M q^{p} .

The constant may depend on the fixed mesh; the estimate is a p-rate on a fixed mesh and makes no claim about h-refinement.

Facts & Assumptions

Given: Countable Choice; a fixed finite matching affine tensor-box mesh; a globally $C^{1}$ function $u$ smooth on each cell; $M > 0$ , $R > 1$ , integer $p \geq 1$ , and $R^{- 1} < q < 1$ .

[F1]

Variable-degree conforming hp spaces on matching affine meshes: $Q_{p} (K)$ is the pullback of degree-at-most- $p$ polynomials in each reference coordinate; a trace-matching box space is defined by equality on shared faces.

[F2]

Endpoint-preserving integrated Legendre projector: the one-coordinate operator is $I_{p} (f (- 1), f^{'}) = f (- 1) + \int_{- 1}^{x} Π_{p - 1} f^{'}$ for an absolutely continuous section with square-integrable derivative.

[F3]

Taylor's Lagrange remainder follows from repeated Rolle steps: for a $C^{p}$ function on an interval containing $0$ and $x$ , the degree- $(p - 1)$ Taylor polynomial at zero has remainder bounded by $\sup ∣ f^{(p)} ∣ ∣ x ∣^{p} / p!$ .

[F4]

Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs: Cauchy–Schwarz holds for the $L^{2}$ inner product on the reference interval and cube.

[F5]

Legendre polynomials have explicit norms and weighted derivative orthogonality: $P_{0} = 1$ and the Legendre family is orthogonal with $\int_{- 1}^{1} P_{0}^{2} = 2$ .

[F6]

The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ : if a function is differentiable on a compact interval and its derivative is Riemann integrable, the integral of that derivative equals the endpoint increment.

[F7]

A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral: under Countable Choice, a bounded Riemann-integrable function has the same Riemann and Lebesgue integrals.

[F8]

Matching variable-degree polynomial traces give an H1 function: under Countable Choice, on a finite matching affine tensor-box mesh, matching polynomial traces give an $H^{1} (Ω)$ function with its cellwise weak gradient.

[A1]

The mixed derivative norm used below is $∥ f ∥_{m i x}^{2} : = \sum_{S \subseteq {1, \dots, d}} ∥ \partial_{S} f ∥_{L^{2} ([- 1, 1]^{d})}^{2}$ , where $\partial_{S}$ differentiates once in each coordinate in $S$ and $\partial_{\emptyset} f = f$ .

Proof

1.1F2F4F5F6F7givenalgebra

On the reference cube define $I_{p}^{(j)}$ by applying [F2] in coordinate $j$ , holding all other coordinates fixed, and put $I_{p} = I_{p}^{(1)} \dots I_{p}^{(d)}$ . Each factor maps degree-at-most- $p$ polynomials in its coordinate to the same degree space, so $I_{p} {\hat{u}}_{K} \in Q_{p}$ . Fix a smooth one-dimensional section $f$ and write $g = f^{'}$ . By [F6], $f (x) - f (- 1)$ is the Riemann integral of $g$ from $- 1$ to $x$ ; [F7] identifies it with the Lebesgue integral used in [F2] and the coefficient formula [F2] (applied to the continuous products $g P_{n}$ ). Orthogonality [F5] makes the constant coefficient exact, so $I_{p} f$ preserves both endpoints. The bound $∣ f (- 1) ∣ \leq 2^{- 1 / 2} ∥ f ∥_{L^{2}} + \sqrt{2} ∥ f^{'} ∥_{L^{2}}$ follows by averaging $f (x) = f (- 1) + \int_{- 1}^{x} f^{'}$ over $x$ and applying Cauchy–Schwarz [F4, F7]; together with $∥ Π_{p - 1} f^{'} ∥_{2} \leq ∥ f^{'} ∥_{2}$ this gives $∥ I_{p} f ∥_{L^{2}} + ∥ (I_{p} f)^{'} ∥_{L^{2}} \leq C ∥ f ∥_{H^{1}}$ uniformly in $p$ . Also, if an absolutely continuous $w$ vanishes at $- 1$ , then $w (x) = \int_{- 1}^{x} w^{'}$ and Cauchy–Schwarz [F4] gives $∥ w ∥_{L^{2}} \leq 2 ∥ w^{'} ∥_{L^{2}}$ .

2.1A1F2F3F6F7step 1.1givenalgebra

Fix $j$ and a subset $S$ of the coordinate indices. If $j \in S$ , then $\partial_{S} (I_{p}^{(j)} - I) {\hat{u}}_{K} = (Π_{p - 1}^{(j)} - I) \partial_{j} \partial_{S ∖ {j}} {\hat{u}}_{K}$ . If $j \notin S$ , then $\partial_{S} (I_{p}^{(j)} - I) {\hat{u}}_{K} = (I_{p}^{(j)} - I) \partial_{S} {\hat{u}}_{K}$ , whose $j$ derivative is $(Π_{p - 1}^{(j)} - I) \partial_{j} \partial_{S} {\hat{u}}_{K}$ and whose value at $- 1$ is zero. Thus in either case the error is controlled by the Legendre projection tail of $g = \partial_{j} \partial_{T} {\hat{u}}_{K}$ , where $T = S ∖ {j}$ in the first case and $T = S$ in the second. Write $r = ∣ T ∣ \leq d - 1$ and let $T_{p - 1}$ be its Taylor polynomial at zero through degree $p - 1$ . Taylor's Lagrange remainder [F3] bounds $∣ g - T_{p - 1} ∣$ on $[- 1, 1]$ by $M (p + r + 1)! R^{- p - r - 1} / p!$ . Orthogonality of $Π_{p - 1}^{(j)}$ gives $∥ g - T_{p - 1} ∥_{2}^{2} = ∥ g - Π_{p - 1}^{(j)} g ∥_{2}^{2} + ∥ Π_{p - 1}^{(j)} g - T_{p - 1} ∥_{2}^{2}$ , so the projection error is at most $\sqrt{2} M (p + r + 1)! R^{- p - r - 1} / p!$ . For fixed $d, R, q$ this is at most $C_{d, R, q} M q^{p}$ : the factorial ratio is bounded by a fixed-degree polynomial in $p$ , while $(R q)^{- p}$ decays geometrically. Anchoring at $- 1$ and the one-dimensional Poincare estimate from step 1.1 use [F6, F7] for the continuous error derivative and give the same bound for each mixed derivative of $(I_{p}^{(j)} - I) {\hat{u}}_{K}$ .

3.1step 1.1step 2.1algebra

The estimate in step 1.1 is stable in every mixed derivative: if a mixed derivative includes coordinate $j$ , differentiating $I_{p}^{(j)}$ there gives the contractive projection $Π_{p - 1}^{(j)}$ ; if it omits $j$ , the one-dimensional $H^{1}$ bound in step 1.1 controls the application of $I_{p}^{(j)}$ using that derivative and its additional $j$ derivative. Summing over the finitely many subsets $S$ gives $∥ I_{p}^{(j)} f ∥_{m i x} \leq C_{d} ∥ f ∥_{m i x}$ uniformly in $p$ . The telescoping identity $I_{p} - I = \sum_{j = 1}^{d} I_{p}^{(1)} \dots I_{p}^{(j - 1)} (I_{p}^{(j)} - I)$ and step 2.1 now imply $∥ {\hat{u}}_{K} - I_{p} {\hat{u}}_{K} ∥_{H^{1} ([- 1, 1]^{d})} \leq C_{d, R, q} M q^{p}$ .

4.1F1F2F8step 3.1givenalgebra∎

The one-dimensional projector preserves both endpoints. It commutes with reflection: both the reflected polynomial and the projection of the reflected function have the same derivative, since reflection preserves the polynomial subspace and the $L^{2}$ inner product, and they have the same value at the left endpoint by preservation of the right endpoint. Coordinate projectors commute under coordinate permutations. On a shared box face the relative affine parameter map is an affine automorphism of the reference cube face: it permutes the supporting facet hyperplanes and preserves their vertices, so its linear part is a signed coordinate permutation. Thus the restrictions of the transported tensor projectors are the same tangential projector applied to the common trace of $u$ , and the resulting cell traces match. The conformity result [F8] gives $u_{p} \in H^{1} (Ω)$ ; since $u \in H^{1} (Ω)$ , the local estimates sum to the stated global $H^{1}$ estimate. Scaling from each reference cell to its physical affine box multiplies the local estimate by a fixed cell-dependent factor; the mesh has finitely many cells, so taking the largest factor gives the global constant $C_{T, d, R, q}$ .

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Constructive varying-degree triangle interpolation in unweighted H1

Statement

Assume Full Choice. Let $T$ be a finite matching shape-regular affine triangulation in two dimensions, let each cell $K$ have an arbitrary finite total degree $p_{K} \geq 1$ , and let $u \in C (\overline{Ω}; R)$ satisfy $u ∣_{K} \in H^{2} (K)$ for every cell. For every edge $E$ , let $τ_{E}$ be either unit tangent direction and write $\partial_{τ_{E}} v = \nabla v \cdot τ_{E}$ . For an interior edge $E = K \cap K^{'}$ , set $p_{E} = \min (p_{K}, p_{K^{'}})$ ; for a boundary edge of $K$ , set $p_{E} = p_{K}$ . Put $m_{K} = \min_{E \subset \partial K} p_{E}$ . There is a linear interpolant $I_{T, p} u$ in the trace-matching space of Variable-degree conforming hp spaces on matching affine meshes. It preserves any zero boundary edge data: if $u$ vanishes on a boundary edge, so does $I_{T, p} u$ . For every $q \in P_{m_{K}} (K; R)$ ,

∣ u - I_{T, p} u ∣_{H^{1} (K)} \leq ∣ u - q ∣_{H^{1} (K)} + C_{γ} \sum_{E \subset \partial K} (ℓ_{E}^{- 1 / 2} ∥ u - q ∥_{L^{2} (E)} + ℓ_{E}^{1 / 2} ∥ \partial_{τ_{E}} (u - q) ∥_{L^{2} (E)}),

and

h_{K}^{- 1} ∥ u - I_{T, p} u ∥_{L^{2} (K)} + ∣ u - I_{T, p} u ∣_{H^{1} (K)} \leq C_{γ} (h_{K}^{- 1} ∥ u - q ∥_{L^{2} (K)} + ∣ u - q ∣_{H^{1} (K)} + \sum_{E \subset \partial K} (ℓ_{E}^{- 1 / 2} ∥ u - q ∥_{L^{2} (E)} + ℓ_{E}^{1 / 2} ∥ \partial_{τ_{E}} (u - q) ∥_{L^{2} (E)})) .

The constant is independent of the degrees and their ratios. In particular, if integer $r \geq 5$ , $m_{K} \geq 2 r - 4$ , $n_{K} = ⌊ m_{K} / 2 ⌋$ , and $u ∣_{K} \in H^{r} (K)$ , then

∣ u - I_{T, p} u ∣_{H^{1} (K)} \leq C_{r, γ} h_{K}^{r - 1} {(\frac{(n_{K} - r + 3)!}{(n_{K} + r - 5)!})}^{1 / 2} ∣ u ∣_{H^{r} (K)} .

For $m_{K} \geq 4 (r - 2)$ the factorial factor is at most $(4 / m_{K})^{r - 4}$ . The effective degree $m_{K}$ records the lowest degree among the shared edge traces; no rate independent of neighboring trace degrees is asserted.

Facts & Assumptions

Given: Full Choice; the finite matching shape-regular affine mesh; finite degrees $p_{K} \geq 1$ ; a real continuous input with cellwise $H^{2}$ regularity; and, for the final rate, integer $r \geq 5$ , $m_{K} \geq 2 r - 4$ , and cellwise $H^{r}$ regularity.

[A1]

Full Choice is assumed through the Sobolev extension and point-evaluation suppliers used below. (The Axiom of Choice)

[F1]

The integrated Legendre operator is $I_{j} f = f (- 1) + \int_{- 1}^{x} Π_{j - 1} f^{'}$ , where $Π_{j - 1}$ is the $L^{2}$ orthogonal projection onto $P_{j - 1}$ ; it is a degree- $j$ polynomial and its derivative projection is a contraction. (Endpoint-preserving integrated Legendre projector)

[F2]

The strong-edge lift constructs a degree-preserving polynomial extension of endpoint-zero edge data, zero on the other two edges, with $H^{1}$ seminorm and $L^{2}$ bounds independent of degree. (A degree-uniform polynomial edge lift with the strong edge norm)

[F3]

The ordinary triangle approximation theorem gives a degree- $p$ polynomial with the displayed scaled $H^{2}$ error and factorial ratio under $H^{r}$ regularity. (Constructive ordinary Sobolev polynomial approximation on triangles)

[F4]

On a triangle, the Sobolev trace maps $H^{2} (K)$ continuously into $H^{1} (E)$ on each edge, and tangential differentiation commutes with the trace; smooth density justifies the derivative trace identity. In dimension two, $H^{2} (K)$ also has a unique continuous representative and well-defined vertex values. (Sobolev face traces and Gauss–Green on a reference simplex, Smooth functions are dense in finite-order Sobolev spaces on a reference simplex, Sobolev threshold for Lagrange point evaluations on a reference simplex, Shape regularity controls affine simplex geometry)

[F5]

Matching polynomial traces on a finite simplicial mesh assemble into an $H^{1}$ function with the cellwise gradient. (Matching variable-degree polynomial traces give an H1 function, Variable-degree conforming hp spaces on matching affine meshes)

[F6]

One-dimensional $H^{1}$ functions have absolutely continuous representatives satisfying the integral formula; Cauchy–Schwarz then gives the endpoint and primitive bounds used for edge values and the zero-boundary polynomial Poincaré estimate. (One-dimensional $W^{1, p}$ functions have unique absolutely continuous representatives, Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs)

Proof

1.1A1F4F6givenconstructalgebra

Justify the Sobolev edge assertion in [F4] from the actual $H^{1}$ trace theorem, rather than assuming a stronger trace statement. On the reference triangle choose smooth $w_{j} \to w$ in $H^{2}$ by its smooth-density supplier. The $H^{1}$ trace theorem applied to $w_{j}$ and separately to each $\partial_{i} w_{j} \in H^{1}$ gives $γ_{E} w_{j} \to γ_{E} w$ and $γ_{E} \nabla w_{j} \to γ_{E} \nabla w$ in $L^{2} (E)$ . For every $ζ \in C_{c}^{\infty} (E)$ , ordinary one-dimensional integration by parts gives $\int_{E} γ_{E} w_{j} \partial_{τ} ζ = - \int_{E} (γ_{E} \nabla w_{j} \cdot τ) ζ$ . Passing to the $L^{2}$ limits proves $\partial_{τ} γ_{E} w = γ_{E} \nabla w \cdot τ$ in distributions and hence $γ_{E} w \in H^{1} (E)$ , with norm at most $C ∥ w ∥_{H^{2}}$ . Its continuous edge representative agrees with the continuous representative of $w$ : smooth approximants converge uniformly on the edge by the one-dimensional $H^{1}$ bound [F6], while the point-evaluation supplier in [F4] gives their convergence to $w$ at each fixed edge point, including the vertices. Transporting the value trace and the gradient trace separately to $K$ yields $∥ w ∥_{L^{2} (E)} \leq C_{γ} (h_{K}^{- 1 / 2} ∥ w ∥_{L^{2} (K)} + h_{K}^{1 / 2} ∣ w ∣_{H^{1} (K)})$ and $∥ \partial_{τ} w ∥_{L^{2} (E)} \leq C_{γ} (h_{K}^{- 1 / 2} ∣ w ∣_{H^{1} (K)} + h_{K}^{1 / 2} ∣ w ∣_{H^{2} (K)})$ . Thus the scaled edge expression in the Statement is bounded by $C_{γ} (h_{K}^{- 1} ∥ w ∥_{2} + ∣ w ∣_{H^{1}} + h_{K} ∣ w ∣_{H^{2}})$ , precisely the scaled $H^{2}$ quantity used in the final rate.

1.2A1F1F2F6givenconstructalgebra

Fix a single orientation of every geometric edge and affine-transport the operator in [F1] to define $g_{E} = I_{p_{E}} (u ∣_{E})$ . For any $s \in P_{p_{E}} (E)$ , its derivative lies in $P_{p_{E} - 1} (E)$ , so the projection is exact and $I_{p_{E}} s = s$ . For a general absolutely continuous $f$ on $E$ , the residual $f^{'} - Π_{p_{E} - 1} f^{'}$ is orthogonal to constants, so its integral over $E$ is zero; the anchor formula and [F6] therefore show that $I_{p_{E}} f$ has both endpoint values of $f$ . In particular $g_{E}$ takes the endpoint values of $u$ . For each cell $K$ , let $a_{K}$ be the affine polynomial with the three vertex values of $u$ . Each residual $d_{E} = g_{E} - a_{K} ∣_{E}$ has zero endpoints and degree at most $p_{E} \leq p_{K}$ . Sum the three lifts from [F2] and put $G_{K} = a_{K} + \sum_{E \subset \partial K} L_{E} d_{E}$ ; it has degree at most $p_{K}$ and trace $g_{E}$ on each edge. Let $Z_{K} = {z \in P_{p_{K}} (K) : z ∣_{\partial K} = 0}$ . In the finite-dimensional affine space $G_{K} + Z_{K}$ , choose the unique $I u ∣_{K}$ satisfying $\int_{K} \nabla (I u - u) \cdot \nabla z = 0$ for every $z \in Z_{K}$ . Existence and uniqueness follow from the positive-definite Gram matrix of the gradients on $Z_{K}$ : if $\int_{K} ∣ \nabla z ∣^{2} = 0$ , the polynomial $z$ is constant, and its zero boundary trace forces $z = 0$ . If $Z_{K} = {0}$ , the system is empty and $I u ∣_{K} = G_{K}$ . Thus the construction is linear and uses only uniquely determined finite systems.

2.1F1F5step 1.2givenalgebra

Both cells incident to an edge use the same oriented edge polynomial $g_{E}$ , so their traces agree and have degree at most $p_{E}$ . By [F5], the assembled interpolant is in the trace-matching $H^{1}$ space. If $E$ is a boundary edge on which $u = 0$ , then $g_{E} = I_{p_{E}} 0 = 0$ , so the interpolant also vanishes there.

2.2F1F2F4F6step 1.2algebra

Fix $q \in P_{m_{K}} (K)$ and put $w = u - q$ . Since $m_{K} \leq p_{E}$ , [F1] reproduces $q ∣_{E}$ , so $g_{E} - q ∣_{E} = I_{p_{E}} (w ∣_{E})$ . Its endpoint values are those of $w$ . The one-dimensional endpoint estimate from [F4] bounds each vertex value $∣ w (v) ∣$ by $C (ℓ_{E}^{- 1 / 2} ∥ w ∥_{L^{2} (E)} + ℓ_{E}^{1 / 2} ∥ \partial_{τ_{E}} w ∥_{L^{2} (E)})$ on any incident edge. Consequently the affine interpolant $a_{K}^{w}$ of the vertex values has $∣ a_{K}^{w} ∣_{H^{1} (K)} + h_{K}^{- 1} ∥ a_{K}^{w} ∥_{L^{2} (K)}$ bounded by the sum of these three edge terms. On each edge, $I_{p_{E}} (w ∣_{E}) - a_{K}^{w} ∣_{E}$ vanishes at its endpoints. The derivative contraction in [F1] and the preceding endpoint estimate bound its strong edge norm by the same sum. Applying [F2] to the three endpoint-zero residuals and adding the lifts gives $c_{K} \in P_{p_{K}} (K)$ with trace $g_{E} - q ∣_{E}$ and $∣ c_{K} ∣_{H^{1} (K)} + h_{K}^{- 1} ∥ c_{K} ∥_{L^{2} (K)} \leq C_{γ} \sum_{E \subset \partial K} (ℓ_{E}^{- 1 / 2} ∥ w ∥_{L^{2} (E)} + ℓ_{E}^{1 / 2} ∥ \partial_{τ_{E}} w ∥_{L^{2} (E)}) .$

3.1F3F4F6step 1.1step 1.2step 2.2algebra∎

The polynomial $q + c_{K}$ has the same boundary trace as $I u ∣_{K}$ , so $z : = q + c_{K} - I u \in Z_{K}$ . The energy orthogonality in step 1.2 makes $I u$ the best $H^{1}$ -seminorm approximation to $u$ in the affine space $G_{K} + Z_{K}$ ; hence $∣ u - I u ∣_{H^{1} (K)} \leq ∣ u - q - c_{K} ∣_{H^{1} (K)} \leq ∣ u - q ∣_{H^{1} (K)} + ∣ c_{K} ∣_{H^{1} (K)}$ . This and step 2.2 prove the first estimate. For the full norm, the reference-triangle identity $z (x, y) = \int_{0}^{x} \partial_{x} z (t, y) d t$ holds because $z = 0$ on $x = 0$ ; Fubini and Cauchy–Schwarz [F6] give $∥ z ∥_{L^{2} (K)} \leq C_{γ} h_{K} ∣ z ∣_{H^{1} (K)}$ . Also $∣ z ∣_{H^{1} (K)} \leq ∣ q + c_{K} - u ∣_{H^{1} (K)} + ∣ u - I u ∣_{H^{1} (K)} \leq 2 (∣ u - q ∣_{H^{1} (K)} + ∣ c_{K} ∣_{H^{1} (K)})$ . Decompose $u - I u = (u - q) - c_{K} + z$ , use the triangle inequality and the $L^{2}$ lift bound from step 2.2, and obtain the displayed full-norm estimate. If $u ∣_{K} \in H^{r} (K)$ , choose $q = A_{m_{K}, r} u$ from [F3]. The proved edge trace bounds in step 1.1 control every edge term by $C_{r, γ} h_{K}^{r - 1} η_{n_{K}, r} ∣ u ∣_{H^{r} (K)}$ , proving the stated rate. Finally $η_{n_{K}, r}^{- 2}$ is the product of $2 (r - 4)$ consecutive integers beginning at $n_{K} - r + 4 \geq m_{K} / 4$ ; taking square roots proves the simplified power bound.

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GLL quadrature changes only the top Legendre modes and is p-uniformly equivalent

Statement

For every $p \geq 1$ , the one-dimensional GLL rule integrates all polynomials of degree at most $2 p - 1$ exactly and has positive weights. If $f = \sum_{n = 0}^{p} a_{n} P_{n}$ , then $Q_{p} (f^{2}) - \int_{- 1}^{1} f^{2} = \frac{2 (p + 1)}{p (2 p + 1)} a_{p}^{2} .$ For every $v \in Q_{p} ([- 1, 1]^{d})$ , $∥ v ∥_{L^{2}}^{2} \leq Q_{p}^{\otimes d} (v^{2}) \leq 3^{d} ∥ v ∥_{L^{2}}^{2}, ∥ \nabla v ∥_{L^{2}}^{2} \leq Q_{p}^{\otimes d} (∣ \nabla v ∣^{2}) \leq 3^{d} ∥ \nabla v ∥_{L^{2}}^{2} .$ The same inequalities hold on affine boxes under the transported rule. If a pointwise-defined scalar coefficient $a : \overline{K} \to R$ is Lebesgue measurable on $K$ and satisfies $0 < \underline{a} \leq a (x) \leq \overline{a}$ throughout $\overline{K}$ , the sampled form $Q_{K, p} (a ∣ \nabla v ∣^{2})$ is equivalent to the exact weighted energy $\int_{K} a ∣ \nabla v ∣^{2}$ with lower factor $\underline{a} / \overline{a}$ and upper factor $3^{d} \overline{a} / \underline{a}$ . These are p-uniform stability bounds; they do not bound the Strang consistency defect or equate quadrature with exact Galerkin assembly.

Facts & Assumptions

Given: Integer $p \geq 1$ , dimension $d \geq 1$ , the GLL nodes and weights, and, for the coefficient bound, a pointwise-defined scalar coefficient $a : \overline{K} \to R$ that is Lebesgue measurable on $K$ and has the stated bounds at every point of $\overline{K}$ .

[F1]

Legendre-Gauss-Lobatto nodes, weights, and tensor quadrature: the nodes are $- 1$ , $1$ , and the roots of $P_{p}^{'}$ , and the weights are interpolatory integrals of the degree- $p$ cardinal polynomials.

[F2]

Legendre polynomials by Rodrigues' formula and their unweighted pairing and Legendre polynomials have explicit norms and weighted derivative orthogonality: $P_{n}$ has degree $n$ , leading coefficient $κ_{n} > 0$ , and $\int_{- 1}^{1} P_{n} P_{m} = 2 δ_{n m} / (2 n + 1)$ .

[A1]

The local tensor polynomial space and affine box conventions are those of Variable-degree conforming hp spaces on matching affine meshes.

Proof

1.1F1F2givenalgebra

Define $g (x) = (x^{2} - 1) P_{p}^{'} (x)$ ; it vanishes at all GLL nodes and has degree $p + 1$ . For $p \geq 2$ , divide any polynomial $f$ of degree at most $2 p - 1$ as $f = g r + s$ with $\deg r \leq p - 2$ and $\deg s \leq p$ . Integration by parts gives $\int_{- 1}^{1} g r = - \int_{- 1}^{1} P_{p} (2 x r + (x^{2} - 1) r^{'}) = 0$ , since the boundary term vanishes and the bracket has degree at most $p - 1$ . The quadrature of $g r$ is also zero because $g$ vanishes at every node, while the interpolatory definition integrates $s$ exactly. Thus $Q_{p} (f) = \int f$ . For $p = 1$ , interpolatory exactness through degree $p = 1$ is the required degree $2 p - 1 = 1$ .

2.1F1F2step 1.1algebra

Divide $P_{p}^{2} = g r + s$ with $\deg r = p - 1$ and $\deg s \leq p$ . Since the leading coefficient of $g$ is $p κ_{p}$ , the leading coefficient of $r$ is $κ_{p} / p$ . Integration by parts as in step 1.1 gives $\int g r = - \int P_{p} (2 x r + (x^{2} - 1) r^{'})$ . The bracket has degree at most $p$ and leading coefficient $(p + 1) κ_{p} / p$ , so it equals $(p + 1) P_{p} / p$ plus a polynomial of degree below $p$ ; orthogonality [F2] therefore gives $\int g r = - (p + 1) ∥ P_{p} ∥_{2}^{2} / p$ . Since $Q_{p} (g r) = 0$ and $Q_{p} (s) = \int s$ , it follows that $Q_{p} (P_{p}^{2}) - \int P_{p}^{2} = - \int g r = (p + 1) ∥ P_{p} ∥_{2}^{2} / p = 2 (p + 1) / (p (2 p + 1))$ .

3.1F1F2step 1.1step 2.1algebra

Write any degree- $p$ polynomial as $f = a_{p} P_{p} + r$ with $\deg r < p$ . Step 1.1 integrates $r^{2}$ and $P_{p} r$ exactly, so step 2.1 gives $Q_{p} (f^{2}) = \int f^{2} + δ_{p} a_{p}^{2}$ , where $δ_{p} = 2 (p + 1) / (p (2 p + 1)) > 0$ . For a cardinal polynomial $ℓ_{j}$ , $Q_{p} (ℓ_{j}^{2}) = w_{j}$ by its nodal values, while the displayed identity gives $w_{j} = \int ℓ_{j}^{2} + δ_{p} a_{p} (ℓ_{j})^{2} > 0$ . Thus all quadrature weights are positive and the one-dimensional squared norm ratio is between $1$ and $2 + 1 / p \leq 3$ .

4.1A1step 1.1step 3.1givenalgebra

Expand a tensor polynomial in the orthogonal basis $\prod_{r = 1}^{d} P_{ν_{r}} (ξ_{r})$ , $0 \leq ν_{r} \leq p$ . The tensor quadrature and exact integral are both diagonal on this basis: every cross term is exact by step 1.1, and each diagonal factor has ratio $1$ unless $ν_{r} = p$ , when step 3.1 bounds it by $3$ . Multiplying the $d$ coordinate factors proves the mass norm bounds. Each coordinate derivative of a tensor polynomial is still in $Q_{p}$ , so applying the same tensor norm bounds to each component and summing proves the gradient bounds.

5.1A1step 4.1givenalgebra∎

An affine change of variables multiplies both exact integrals and the transported quadrature by the same positive Jacobian. Apply the unweighted bounds of step 4.1 to each physical gradient component pulled back to the reference cube. The measurable bounded coefficient makes $a ∣ \nabla v ∣^{2}$ integrable on the bounded box, and its specified values on $\overline{K}$ define every quadrature sample. Positivity of the weights and the pointwise bounds give $\underline{a} Q (∣ \nabla v ∣^{2}) \leq Q (a ∣ \nabla v ∣^{2}) \leq \overline{a} Q (∣ \nabla v ∣^{2})$ and the analogous inequalities for the exact integral; combine them with step 4.1 to obtain the stated weighted comparison. The constants are independent of $p$ .

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Exponential hp energy convergence for weighted-analytic polygon corners

Statement

Assume Countable Choice. Let $Ω \subset R^{2}$ be a bounded polygonal open domain with finite corner set $C = {c_{1}, \dots, c_{m}}$ and weights $0 \leq β_{i} < 1$ . For $γ \in [0, \infty)^{m}$ define $Φ_{γ} (x) = \prod_{i} ∣ x - c_{i} ∣^{γ_{i}}$ ; $β + j - 2$ means componentwise addition. For integer $k \geq 2$ , define

∥ v ∥_{H_{β}^{k, 2} (Ω)}^{2} = ∥ v ∥_{H^{1} (Ω)}^{2} + \sum_{j = 2}^{k} \sum_{∣ α ∣ = j} ∥ Φ_{β + j - 2} D^{α} v ∥_{2}^{2} .

Let $B_{β}^{2} (Ω)$ consist of real-valued functions in every $H_{β}^{k, 2} (Ω)$ for which some $C_{u}, d_{u} > 0$ satisfy

{∥ Φ_{β + k - 2} \sum_{∣ α ∣ = k} ∣ D^{α} u ∣ ∥}_{2} \leq C_{u} d_{u}^{k - 2} (k - 2)! (k \geq 2) .

Let $(T_{k})_{k \geq 1}$ be nested finite matching shape-regular affine triangulations of $Ω$ , with at most $C_{0} k$ triangles. Write $h_{T} = diam (T)$ and $dist (T, C) = \inf_{x \in T, c \in C} ∣ x - c ∣$ . Assume a fixed $0 < σ < 1$ and $C_{1} > 0$ satisfy

σ \leq \frac{h_{T}}{dist (T, C)} \leq σ^{- 1} (\overline{T} \cap C = \emptyset), h_{T} \leq C_{1} σ^{k} (\overline{T} \cap C \neq \emptyset) .

Let $p_{k} \in N$ be uniform total polynomial degrees with $c_{0} k \leq p_{k} \leq c_{1} k$ for fixed $0 < c_{0} \leq c_{1}$ . For a fixed union $Γ_{D}$ of polygon edges, let $V_{k}$ be the space of broken piecewise- $P_{p_{k}}$ functions whose polynomial traces agree on every interior edge and vanish on the boundary edges in $Γ_{D}$ ; put $N_{k} = \dim V_{k}$ . This is the corresponding subspace of the variable-degree space in Variable-degree conforming hp spaces on matching affine meshes. For this smoother class, the boundary condition means that the unique continuous representative supplied by Weighted analytic polygon functions have corner decay and scaled analytic bounds vanishes pointwise on the edges in $Γ_{D}$ ; no zero condition is imposed on the other edges. If $u \in B_{β}^{2} (Ω)$ satisfies this boundary condition, then constants $C, b > 0$ , independent of $k$ , satisfy

\inf_{v_{k} \in V_{k}} ∥ \nabla (u - v_{k}) ∥_{L^{2} (Ω)} \leq C e^{- b N_{k}^{1 / 3}} .

For a bounded coercive conforming Galerkin problem on a Hilbert space $V \subset H^{1} (Ω)$ , assume $u \in V$ is its exact variational solution: $a (u, v) = F (v)$ for every $v \in V$ , with bounded load $F$ . If $V_{k} \subset V$ and the norm on $V$ is equivalent to the gradient seminorm, Céa's estimate gives the same exponential rate for the energy error, with its actual $M / α$ and norm-equivalence factors included. The constants may depend on the polygon, weights, weighted analytic bounds, mesh shape and grading constants, degree constants, $Γ_{D}$ , and the Galerkin coercivity and norm-equivalence constants.

Facts & Assumptions

Given: Countable Choice; the polygon, corners and weights; the full weighted-analytic hypotheses; a nested shape-regular matching mesh sequence with the exact grading and count assumptions; uniform total degrees $c_{0} k \leq p_{k} \leq c_{1} k$ ; and the fixed boundary-edge set $Γ_{D}$ .

[A1]

Countable Choice is assumed explicitly; no full Choice is used in this proof. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

The weighted corner estimates give a continuous representative, corner decay $∣ u - u (c_{i}) ∣ \leq C r^{a}$ , pointwise derivative bounds, and analytic scaled bounds on regular triangles. (Weighted analytic polygon functions have corner decay and scaled analytic bounds)

[F2]

The analytic polynomial supplier gives exponential $W^{1, \infty}$ approximation on a reference triangle and an endpoint-preserving interval projector whose derivative is an $L^{2}$ contraction with exponential $H^{1}$ error. (Analytic derivative bounds give exponential polynomial approximation on a triangle)

[F3]

The degree-uniform strong-edge lift extends an endpoint-zero polynomial from one edge, vanishes on the other edges, preserves degree, and has seminorm at most $C ℓ_{E}^{1 / 2} ∥ \partial_{τ} q ∥_{2}$ . (A degree-uniform polynomial edge lift with the strong edge norm)

[F4]

The variable-degree space is the finite matching polynomial space with trace equality; the conformity lemma glues matching polynomial traces into $H^{1}$ ; and the corner count gives $N_{k} \leq C_{N} k^{3}$ . (Variable-degree conforming hp spaces on matching affine meshes, Matching variable-degree polynomial traces give an H1 function, Uniform-degree polygon corner spaces have cubic degree-of-freedom growth)

[F5]

Céa's theorem gives $∥ u - u_{k} ∥_{V} \leq (M / α) \inf_{v_{k} \in V_{k}} ∥ u - v_{k} ∥_{V}$ for the exact variational solution $u$ and its Galerkin solution $u_{k}$ , under its stated bounded-load, boundedness, coercivity, conformity, and Countable Choice hypotheses. (Céa quasi-optimality for coercive conforming Galerkin problems)

[F6]

Cauchy–Schwarz controls one-dimensional trace errors and the area estimates on the regular triangles. (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs)

Proof

1.1A1F1givenalgebra

Fix $0 < a < \min_{i} (1 - β_{i})$ as in [F1]. Call a cell a cap when its closure contains a corner. Since the corner set is finite and distinct corners have positive separation, for all sufficiently large $k$ each cap contains at most one corner; its diameter is at most $C_{1} σ^{k}$ . On a noncap cell $T$ with $d_{T} = dist (T, C) < R$ , the mesh condition gives $σ \leq h_{T} / d_{T} \leq σ^{- 1}$ , so [F1] gives uniform scaled analytic derivative bounds after subtracting a suitable constant. If $d_{T} \geq R$ , the compact-away-from-corners bounds in [F1], together with bounded $h_{T}$ and the fixed $r_{T} = R$ , give the same conclusion after enlarging constants. Thus each noncap reference pullback has derivative bounds $C r_{T}^{a} A_{1}^{∣ α ∣} ∣ α ∣!$ , where $r_{T} = \min (R, d_{T})$ .

2.1F1F2F6step 1.1givenconstruct

For every mesh edge incident to a polygon corner, prescribe the linear interpolant of the two endpoint values of the continuous representative of $u$ . All cells sharing such an edge are caps. On every other edge $E$ , prescribe the single common endpoint-preserving derivative-projection polynomial $g_{E}$ from [F2], in one globally fixed edge orientation. The edge lies a distance comparable to its length from any nearby corner: for an edge opposite a cap corner this follows from shape regularity and the altitude bound, and for a noncap neighbor it also follows from the mesh ratio condition. Thus [F1]–[F2] give, on each noncap edge, $∥ u - g_{E} ∥_{L^{\infty} (E)} + ℓ_{E}^{1 / 2} ∥ \partial_{τ} (u - g_{E}) ∥_{L^{2} (E)} \leq C r_{T}^{a} e^{- b p_{k}}$ for each adjoining noncap cell, with comparable scales on both sides. Here the sup bound follows from the endpoint identity and $∥ e ∥_{\infty} \leq ∥ e^{'} ∥_{L^{1}} \leq ∥ e^{'} ∥_{L^{2}}$ for the endpoint-zero reference error; [F6] justifies the last inequality. On an edge incident to a corner, both adjacent cells are caps, its prescribed linear polynomial shares the exact continuous vertex values, and no tangential regularity of $u$ at the corner is used. Every edge polynomial takes the value of $u$ at both endpoints. If $E \subset Γ_{D}$ , these endpoint values and the entire prescribed edge polynomial are zero.

3.1F2F3step 1.1step 2.1algebra

Let $T$ be a noncap triangle and choose $q_{T} \in P_{p_{k}} (T)$ by transporting the analytic approximation [F2] for the scaled pullback, then adding back its subtracted constant. Thus $∥ u - q_{T} ∥_{\infty} + h_{T} ∥ \nabla (u - q_{T}) ∥_{\infty} \leq C r_{T}^{a} e^{- b p_{k}}$ . Correct its three vertex values by an affine polynomial to obtain ${\tilde{q}}_{T}$ matching the prescribed $g_{E}$ at all vertices; the correction has $H^{1}$ seminorm at most $C r_{T}^{a} e^{- b p_{k}}$ . On each edge, $g_{E} - {\tilde{q}}_{T} ∣_{E}$ has zero endpoints, degree at most $p_{k}$ , and strong edge derivative norm at most $C r_{T}^{a} e^{- b p_{k}}$ . Lift these three residuals separately by [F3]. Their sum does not change the other edge traces, so the resulting $v_{T} \in P_{p_{k}} (T)$ has all three prescribed traces and $∣ u - v_{T} ∣_{H^{1} (T)} \leq C r_{T}^{a} e^{- b p_{k}} \leq C e^{- b p_{k}} .$

3.2F1F2F3F6step 1.1step 2.1algebra

Let $T$ be a cap at corner $c_{i}$ , with diameter $h_{T}$ ; for large $k$ it contains no other corner. Start with the affine interpolant $a_{T}$ of the three exact vertex values. By [F1], $∣ u - a_{T} ∣_{H^{1} (T)} \leq C h_{T}^{a}$ . Its two edges incident to $c_{i}$ already have the prescribed linear traces. On the opposite edge $E$ , every point has distance comparable to $h_{T}$ from $c_{i}$ , so the gradient bound in [F1] and Cauchy–Schwarz [F6] give $ℓ_{E}^{1 / 2} ∥ \partial_{τ} u ∥_{L^{2} (E)} \leq C h_{T}^{a}$ . The interval derivative projection in [F2] is contractive, and the vertex estimate in [F1] gives the same bound for $g_{E} - a_{T} ∣_{E}$ . This discrepancy vanishes at both endpoints; add its lift from [F3]. The resulting polynomial has all prescribed traces and $∣ u - v_{T} ∣_{H^{1} (T)} \leq C h_{T}^{a} \leq C σ^{a k}$ . No $H^{2}$ regularity is used on a cap.

4.1F4step 3.1step 3.2givenalgebra

Every pair of adjacent local polynomials has common trace $g_{E}$ ; the boundary traces vanish on $Γ_{D}$ . Thus they assemble to $v_{k} \in V_{k}$ by [F4]. Summing squared cell errors from steps 3.1 and 3.2 over at most $C_{0} k$ cells gives $∥ \nabla (u - v_{k}) ∥_{2} \leq C \sqrt{k} (e^{- b p_{k}} + σ^{a k}) \leq C^{'} e^{- η k}$ for some $η > 0$ , because $p_{k} \geq c_{0} k$ and $0 < σ < 1$ . The finitely many earlier meshes are handled by $v_{k} = 0$ and an increase of $C^{'}$ . Since $N_{k} \leq C_{N} k^{3}$ by [F4], $k \geq C_{N}^{- 1 / 3} N_{k}^{1 / 3}$ and therefore $\inf_{v \in V_{k}} ∥ \nabla (u - v) ∥_{2} \leq C e^{- b N_{k}^{1 / 3}}$ .

5.1F5step 4.1givenalgebra∎

Under the additional Galerkin hypotheses, $u$ satisfies $a (u, v) = F (v)$ for every $v \in V$ , and the Galerkin solution $u_{k} \in V_{k}$ satisfies $a (u_{k}, v_{k}) = F (v_{k})$ for every $v_{k} \in V_{k}$ . Thus Céa [F5] applies to this same $u$ and $u_{k}$ . Apply it to the best-approximation estimate of step 4.1. Norm equivalence between $∥ \cdot ∥_{V}$ and the gradient seminorm changes only the stated constant, and the actual $M / α$ factor is retained. This proves the energy-error conclusion.

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Cea transfers the fixed-mesh analytic p rate to the energy error

Statement

Assume Countable Choice. On a fixed finite matching affine tensor-box mesh, let $V = H^{1} (Ω)$ with its standard norm, let $a (v, w) = \int_{Ω} (\nabla v \cdot \nabla w + v w)$ , and let $F$ be a bounded linear functional on $V$ . Let $u \in V$ solve $a (u, w) = F (w)$ for all $w \in V$ , and let $u_{p}$ be its conforming Galerkin solution in the common-degree trace-matching tensor space of Variable-degree conforming hp spaces on matching affine meshes. If the analytic hypotheses of Analytic data converge exponentially under fixed-mesh p refinement hold with constants $M, R$ and $R^{- 1} < q < 1$ , then

∥ u - u_{p} ∥_{H^{1} (Ω)} \leq C_{T, d, R, q} M q^{p} .

This is the fixed-mesh analytic branch; it does not assert an h-dependent estimate or an ordinary Sobolev p-rate on simplices.

Facts & Assumptions

Given: Countable Choice; the finite matching affine tensor mesh; $a, F, u, V$ as in the Statement; integer $p \geq 1$ ; and the analytic derivative bound and $q$ of the fixed-mesh projector theorem.

[F1]

Matching variable-degree polynomial traces give an H1 function: the trace-matching tensor space is a finite-dimensional subspace of $H^{1} (Ω)$ .

[F2]

Analytic data converge exponentially under fixed-mesh p refinement: there is a trace-matching $u_{p}^{*} \in V_{p}$ with $∥ u - u_{p}^{*} ∥_{H^{1}} \leq C_{T, d, R, q} M q^{p}$ .

[F3]

Céa quasi-optimality for coercive conforming Galerkin problems: under Countable Choice, a bounded coercive form on a Hilbert space and a finite-dimensional conforming subspace have a Galerkin solution satisfying $∥ u - u_{p} ∥_{V} \leq (M_{a} / α_{a}) \inf_{v_{p} \in V_{p}} ∥ u - v_{p} ∥_{V}$ .

[A1]

For the form in the Statement, Cauchy–Schwarz gives $∣ a (v, w) ∣ \leq ∥ v ∥_{H^{1}} ∥ w ∥_{H^{1}}$ and $a (v, v) = ∥ v ∥_{H^{1}}^{2}$ .

Proof

1.1A1F1F3given

By [F1], $V_{p}$ is finite-dimensional and contained in the Hilbert space $V$ . The form in [A1] is bounded with $M_{a} = 1$ and coercive with $α_{a} = 1$ . The exact solution and bounded functional are among the hypotheses in Given, so [F3] applies and yields $∥ u - u_{p} ∥_{H^{1}} \leq \inf_{v_{p} \in V_{p}} ∥ u - v_{p} ∥_{H^{1}}$ . Its only axiom beyond classical real analysis is Countable Choice, stated in Given.

2.1F2step 1.1algebra∎

The approximant $u_{p}^{*}$ in [F2] belongs to $V_{p}$ , so the infimum in step 1.1 is no greater than $∥ u - u_{p}^{*} ∥_{H^{1}}$ . Substituting its bound from [F2] proves the stated energy estimate.

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Galerkin energy error with local h, trace degree and Sobolev regularity

Statement

Assume the Axiom of Choice. Let $T$ be any finite matching shape-regular affine triangulation of a bounded polygonal open set $Ω \subset R^{2}$ , with shape bound $γ$ and arbitrary degrees $p_{K} \geq 1$ . Set $p_{E} = \min (p_{K}, p_{K^{'}})$ on an interior edge, $p_{E} = p_{K}$ on a boundary edge, and $m_{K} = \min_{E \subset \partial K} p_{E}$ . Thus the effective degree retains neighboring shared-trace restrictions, without a bound on degree ratios.

Choose $χ = 1$ and $V = H^{1} (Ω)$ , or $χ = 0$ and $V = H_{0}^{1} (Ω)$ , with Hilbert norm $∥ v ∥_{V}^{2} = ∣ v ∣_{H^{1} (Ω)}^{2} + χ ∥ v ∥_{L^{2} (Ω)}^{2} .$ Let $V_{T, p}$ be the corresponding conforming hp space, imposing zero boundary traces in the $χ = 0$ branch. Suppose the real bilinear form $a$ and real linear functional $F$ satisfy $∣ a (v, w) ∣ \leq M_{a} ∥ v ∥_{V} ∥ w ∥_{V}, a (v, v) \geq α_{a} ∥ v ∥_{V}^{2}, α_{a} > 0,$ with $F$ bounded. Let $u \in V$ be the variational solution and $u_{T, p}$ its conforming Galerkin solution. Suppose $u \in C (\overline{Ω})$ and $u ∣_{K} \in H^{2} (K)$ ; in the zero-boundary branch its continuous representative vanishes on the boundary edges.

For any cellwise comparison polynomials $q_{K} \in P_{m_{K}} (K)$ put $w_{K} = u - q_{K}$ and $J_{K} (q_{K}) = ∣ w_{K} ∣_{H^{1} (K)} + \sum_{E \subset \partial K} (ℓ_{E}^{- 1 / 2} ∥ w_{K} ∥_{L^{2} (E)} + ℓ_{E}^{1 / 2} ∥ \partial_{τ_{E}} w_{K} ∥_{L^{2} (E)}), R_{K} (q_{K}) = h_{K}^{- 1} ∥ w_{K} ∥_{L^{2} (K)} + J_{K} (q_{K}) .$ Then the actual energy-space error obeys $∥ u - u_{T, p} ∥_{V} \leq \frac{M_{a}}{α_{a}} C_{γ} {(\sum_{K \in T} [J_{K} (q_{K})^{2} + χ h_{K}^{2} R_{K} (q_{K})^{2}])}^{1 / 2} .$ This comparison estimate applies to every degree assignment and the stated piecewise $H^{2}$ inputs; the comparison polynomials need not match across edges.

For any cellwise integers $s_{K} \geq 2$ with $m_{K} \geq s_{K} - 1$ and $u ∣_{K} \in H^{s_{K}} (K)$ , this also gives the general h-refinement bound $∥ u - u_{T, p} ∥_{V} \leq \frac{M_{a}}{α_{a}} {(\sum_{K} C_{s_{K}, γ}^{2} (1 + χ h_{K}^{2}) h_{K}^{2 (s_{K} - 1)} ∣ u ∣_{H^{s_{K}} (K)}^{2})}^{1 / 2} .$ In particular every assignment $p_{K} \geq 1$ admits the piecewise $H^{2}$ estimate with $s_{K} = 2$ .

For the additional explicit p-dependent refinement, let integers $r_{K} \geq 5$ satisfy $m_{K} \geq 2 r_{K} - 4$ , assume $u ∣_{K} \in H^{r_{K}} (K)$ , and put $n_{K} = ⌊ m_{K} / 2 ⌋$ and $η_{K} = {(\frac{(n_{K} - r_{K} + 3)!}{(n_{K} + r_{K} - 5)!})}^{1 / 2} .$ There are constants $C_{r_{K}, γ}$ independent of all diameters, degrees and degree ratios such that $∥ u - u_{T, p} ∥_{V} \leq \frac{M_{a}}{α_{a}} {(\sum_{K \in T} C_{r_{K}, γ}^{2} (1 + χ h_{K}^{2}) h_{K}^{2 (r_{K} - 1)} η_{K}^{2} ∣ u ∣_{H^{r_{K}} (K)}^{2})}^{1 / 2} .$ If $m_{K} \geq 4 (r_{K} - 2)$ , one may replace $η_{K}$ by $(4 / m_{K})^{r_{K} - 4}$ . For a common regularity $r$ , this displays local $h_{K}^{r - 1}$ and the proved conservative trace-degree factor $m_{K}^{- (r - 4)}$ . It does not claim the optimal $p^{- (r - 1)}$ rate or any bound on growth of $C_{r, γ}$ as $r \to \infty$ . The separate fixed-mesh tensor-analytic and weighted-corner exponential results remain independent branches.

Facts & Assumptions

Given: AC; the finite matching shape-regular two-dimensional mesh, degrees and either space/norm branch; the actual boundedness and coercivity constants; the exact solution with the stated continuity and cellwise regularity; and the comparison polynomials or the additional ordinary Sobolev assumptions.

[F1]

Constructive varying-degree triangle interpolation in unweighted H1 gives a single linear trace-matching interpolant $I u$ , preserving zero boundary edge values, with $∣ u - I u ∣_{H^{1} (K)} \leq C_{γ} J_{K} (q_{K})$ and $h_{K}^{- 1} ∥ u - I u ∥_{L^{2} (K)} + ∣ u - I u ∣_{H^{1} (K)} \leq C_{γ} R_{K} (q_{K})$ . Its local $H^{2}$ edge trace and tangential-derivative identity are proved there from smooth density and the actual first-order trace theorem.

[F2]

Constructive ordinary Sobolev polynomial approximation on triangles supplies $q_{K} = A_{K, m_{K}, r_{K}} u$ with $(\sum_{j = 0}^{2} h_{K}^{2 j - 2} ∣ u - q_{K} ∣_{H^{j} (K)}^{2})^{1 / 2} \leq C_{r_{K}, γ} h_{K}^{r_{K} - 1} η_{K} ∣ u ∣_{H^{r_{K}} (K)}$ , and the stated high-degree simplification.

[F3]

Matching variable-degree polynomial traces give an H1 function and Variable-degree conforming hp spaces on matching affine meshes give a finite-dimensional conforming space and identify its weak gradients with its cellwise gradients.

[F4]

Céa quasi-optimality for coercive conforming Galerkin problems gives exact and discrete solvability and the factor $M_{a} / α_{a}$ under Countable Choice. Integer-order Sobolev spaces are Banach, Zero-boundary Sobolev space as a norm closure and The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction give completeness of the two specified Hilbert norms; AC supplies their choice assumptions.

[F5]

Bramble–Hilbert approximation on a reference simplex with Sobolev exponent two and $r = s_{K} \geq 2$ supplies a degree- $(s_{K} - 1)$ polynomial with reference $H^{2}$ error bounded by the $H^{s_{K}}$ seminorm. Shape regularity controls affine simplex geometry transports that estimate with the exact factors $h_{K}^{j - 1}$ , $j = 0, 1, 2$ .

[F6]

Zero-boundary conforming polynomials lie in W0 1p and satisfy Poincare: on a finite matching simplicial mesh, a conforming piecewise-polynomial function with zero polynomial trace on every boundary facet belongs to $H_{0}^{1} (Ω)$ .

Proof

1.1F3F4given

Both norm branches are Hilbert: the Sobolev inner product is complete by [F4], the zero-boundary Sobolev closure is closed, and Poincaré makes its gradient norm equivalent to the inherited complete norm. The finite matching polynomial space is finite-dimensional and conforming by [F3]. The given exact form and load therefore meet Céa's precise hypotheses, giving unique variational solutions and $∥ u - u_{T, p} ∥_{V} \leq (M_{a} / α_{a}) \inf_{v \in V_{T, p}} ∥ u - v ∥_{V}$ .

1.2F1F3F4F6constructalgebra

The interpolant [F1] belongs to the trace-matching space. In the zero-boundary branch, [F1] also gives zero polynomial traces on every boundary facet. Apply [F6] with exponent $2$ to obtain $I u \in H_{0}^{1} (Ω)$ , so the interpolant is an admissible comparison in either norm branch. No neighboring degree ratio was used.

2.1F1F3step 1.1step 1.2algebra

Sum the squared cell seminorm bounds in [F1], which gives $∣ u - I u ∣_{H^{1} (Ω)}^{2} \leq C_{γ}^{2} \sum_{K} J_{K} (q_{K})^{2}$ by [F3]. The full scaled estimate also gives $∥ u - I u ∥_{L^{2} (Ω)}^{2} \leq C_{γ}^{2} \sum_{K} h_{K}^{2} R_{K} (q_{K})^{2}$ . Their sum with the exact factor $χ$ bounds $∥ u - I u ∥_{V}^{2}$ . Use $I u$ as the admissible comparison in step 1.1 and take square roots to obtain the first displayed Galerkin bound. The local comparisons may be chosen independently because the single interpolant was constructed from common edge traces of $u$ , rather than by gluing the $q_{K}$ .

3.1F1F5step 2.1algebra

Choose the reference moment polynomial of degree $s_{K} - 1$ from [F5] and transport it to $K$ . The degree condition makes it an admissible $q_{K} \in P_{m_{K}} (K)$ . With $w_{K} = u - q_{K}$ , scaling gives $h_{K}^{- 1} ∥ w_{K} ∥_{2} + ∣ w_{K} ∣_{H^{1}} + h_{K} ∣ w_{K} ∣_{H^{2}} \leq C_{s_{K}, γ} h_{K}^{s_{K} - 1} ∣ u ∣_{H^{s_{K}} (K)}$ . The value and derivative edge trace bounds proved in [F1] therefore give the same bound for $J_{K}$ and $R_{K}$ . Substitute into step 2.1 to prove the general integer-regularity h estimate, including $s_{K} = 2$ on every degree assignment.

4.1F1F2step 2.1algebra∎

Under the stronger cellwise assumptions choose $q_{K}$ from [F2]. The scaled edge trace proved in [F1] gives $J_{K} (q_{K}), R_{K} (q_{K}) \leq C_{r_{K}, γ} h_{K}^{r_{K} - 1} η_{K} ∣ u ∣_{H^{r_{K}} (K)}$ : its value-edge term is bounded by $C (h_{K}^{- 1} ∥ w_{K} ∥_{2} + ∣ w_{K} ∣_{H^{1}})$ , and its derivative-edge term by $C (∣ w_{K} ∣_{H^{1}} + h_{K} ∣ w_{K} ∣_{H^{2}})$ , so exactly the three scaled norms in [F2] occur. Substitution into step 2.1 proves the second displayed bound with $(1 + χ h_{K}^{2})$ . Finally [F2] gives $η_{K} \leq (4 / m_{K})^{r_{K} - 4}$ under the stronger degree threshold. This retains the actual trace-degree and regularity constants and concludes both norm branches.

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GLL spectral-element energy error retains the quadrature defect

Statement

Assume Countable Choice. Let $Ω$ have a fixed finite matching affine tensor-box mesh with cell degrees $p_{K} \geq 1$ , and let $V_{p}$ be its trace-matching hp space. On $V = H^{1} (Ω)$ use $a (v, w) = \int_{Ω} (\nabla v \cdot \nabla w + v w)$ and a bounded load $F$ . Define the discrete form on $V_{p}$ by

a_{p} (v_{p}, w_{p}) = \sum_{K \in T} (a_{K, p_{K}} (v_{p} ∣_{K}, w_{p} ∣_{K}) + m_{K, p_{K}} (v_{p} ∣_{K}, w_{p} ∣_{K})),

using the GLL stiffness and mass forms from Legendre-Gauss-Lobatto nodes, weights, and tensor quadrature. Let $u$ solve the exact variational problem and let $u_{p} \in V_{p}$ solve $a_{p} (u_{p}, w_{p}) = F (w_{p})$ for every $w_{p} \in V_{p}$ , with the exact load restricted to $V_{p}$ . Define the form defect $δ_{a, p}$ by Variational-crime settings and consistency defects. Then $∥ u - u_{p} ∥_{H^{1} (Ω)} \leq \inf_{v_{p} \in V_{p}} (2 ∥ u - v_{p} ∥_{H^{1} (Ω)} + δ_{a, p} (v_{p})) .$ The stability factor is uniform in all cell degrees. The defect is retained as stated; no aliasing or p-decay estimate is asserted, and GLL assembly is not identified with exact Galerkin assembly.

Facts & Assumptions

Given: Countable Choice; the finite matching affine tensor-box mesh and $p_{K} \geq 1$ ; exact $a, F, u$ ; the GLL form $a_{p}$ ; and its discrete solution $u_{p}$ .

[F1]

Matching variable-degree polynomial traces give an H1 function: the trace-matching hp space is finite-dimensional and contained in $H^{1} (Ω)$ .

[F2]

GLL quadrature changes only the top Legendre modes and is p-uniformly equivalent: on each affine box the GLL quadrature H1 form is bounded above by $3^{d}$ times and below by $1$ times the exact local H1 form, uniformly in its degree.

[F3]

Variational-crime settings and consistency defects: $δ_{a, p} (v_{p}) = \sup_{w_{p} \neq 0} ∣ a (v_{p}, w_{p}) - a_{p} (v_{p}, w_{p}) ∣ / ∥ w_{p} ∥_{H^{1}}$ , while the load defect is defined analogously.

[F4]

First Strang estimate for conforming perturbed Galerkin problems: under Countable Choice, a coercive exact and discrete form on a finite-dimensional conforming subspace satisfy the first Strang estimate with the form and load defects from [F3].

[F5]

Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs: a positive weighted finite sum obeys Cauchy–Schwarz.

[F6]

Legendre-Gauss-Lobatto nodes, weights, and tensor quadrature defines the cell stiffness form $a_{K, p_{K}}$ and mass form $m_{K, p_{K}}$ used in the displayed discrete form.

Proof

1.1F1F2F3F5F6givenalgebra

Sum [F2] over cells. For every $v_{p} \in V_{p}$ , $a (v_{p}, v_{p}) = ∥ v_{p} ∥_{H^{1}}^{2}$ and $a_{p} (v_{p}, v_{p}) \geq ∥ v_{p} ∥_{H^{1}}^{2}$ ; Cauchy–Schwarz [F5] for the positive quadrature weights gives $∣ a_{p} (v_{p}, w_{p}) ∣ \leq 3^{d} ∥ v_{p} ∥_{H^{1}} ∥ w_{p} ∥_{H^{1}}$ . Thus the discrete form is bounded and coercive with $α_{p} \geq 1$ , and the exact form has $M = 1$ . The exact load is unchanged on $V_{p}$ , so $δ_{F, p} = 0$ .

2.1F1F3F4step 1.1givenalgebra∎

Apply [F4] with $V = H^{1} (Ω)$ , the finite-dimensional conforming subspace from [F1], exact bounded coercive form $a$ , GLL form $a_{p}$ , and the restricted load. Its bound is $\inf_{v_{p} \in V_{p}} ((1 + M / α_{p}) ∥ u - v_{p} ∥ + (δ_{a, p} (v_{p}) + δ_{F, p}) / α_{p})$ . Substituting $M = 1$ , $α_{p} \geq 1$ , and $δ_{F, p} = 0$ from step 1.1 yields the stated inequality. Countable Choice is the only axiom used by [F4].

5 · Examples, counterexamples and false statements

None yet.