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Finite-element meshes and interpolation

1 · Prerequisites

2 · Summary

A finite face-to-face affine simplicial mesh is the setting for the interpolation and differential identities on this page. The construction fixes an orientation-preserving reference map and the diameter-to-inradius shape-regularity convention; curved cells are outside this scope. Oriented simplex chains, matching-facet cancellation, affine-simplex Gauss–Green, Sobolev extension, face traces, and Poincare–Wirtinger are established locally where the bounded C1-domain suppliers do not apply.

For conforming piecewise polynomials, equal interior traces give the weak cellwise gradient, while zero boundary facet values yield membership in the W0 closure and its Poincare estimate. Reference finite elements are stated conditionally on unisolvent, bounded degrees of freedom. Bramble–Hilbert and affine scaling give local interpolation, inverse, and discrete-trace estimates. Point-evaluation boundedness is proved for $ℓ > d / p$ when $p < \infty$ , and for $p = \infty$ . The two-dimensional $W^{1, 2}$ counterexample proves unboundedness at that critical case; no other threshold cases are characterized.

The covariant and contravariant Piola maps are defined with det A>0. Their weak gradient, divergence and curl identities and normal/tangential moment transformations are proved on affine simplices. Transfer of a commuting interpolation diagram is conditional on the corresponding reference diagram, so arbitrary unisolvent elements are not asserted to commute.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

Matching affine simplicial meshes and shape regularity

Definition

Fix an integer $d \geq 2$ . The underlying coordinate index set of $R^{d}$ is $I_{d} = {0, \dots, d - 1}$ . In this definition the conventional labels $1 \leq i \leq d$ are relabeled by $i \mapsto i - 1$ : write ${\hat{x}}_{i} : = \hat{x} (i - 1)$ , and let $e_{i} \in R^{d}$ be the function on $I_{d}$ with $e_{i} (i - 1) = 1$ and $e_{i} (k) = 0$ for $k \neq i - 1$ . The vertex and barycentric labels $0, \dots, d$ are distinct from these underlying coordinate indices. The reference open simplex is ${\hat{S}}_{d} : = {\hat{x} \in R^{d} : {\hat{x}}_{i} > 0 (1 \leq i \leq d), \sum_{i = 1}^{d} {\hat{x}}_{i} < 1},$ with closure $\overline{{\hat{S}}_{d}} = conv {0, e_{1}, \dots, e_{d}}$ .

A matching affine simplicial mesh of a bounded polyhedral open set $Ω \subset R^{d}$ is a finite set $T$ of closed, nondegenerate $d$ -simplices such that their union is $\overline{Ω}$ , their interiors are pairwise disjoint, $Ω$ is the interior of their union, and the intersection of two distinct cells is either empty or a whole face of each. Thus no lower-dimensional crack is omitted from $Ω$ while its surrounding cells are retained. Every interior facet is incident to exactly two cells and every boundary facet to one. The faces of a cell are the convex hulls of its vertex subsets. A facet means a face of dimension $d - 1$ .

For each cell $K \in T$ , an affine bijection $F_{K} : \overline{{\hat{S}}_{d}} \to K, F_{K} (\hat{x}) = b_{K} + A_{K} \hat{x},$ with invertible $A_{K} \in R^{d \times d}$ is a reference map. Choose the vertex order so $\det A_{K} > 0$ ; this is the orientation convention used for Piola formulas. Write $h_{K} : = diam (K)$ and let $ρ_{K}$ be the radius of the largest Euclidean ball contained in $K$ . A family ${T_{h}}_{h \in H}$ is shape regular if there is a finite $σ$ such that $h_{K} / ρ_{K} \leq σ$ for every $h \in H$ and $K \in T_{h}$ . This universal condition is vacuous for an empty family. This definition uses radius; references using the diameter of the inscribed ball differ by a factor of two.

An affine image of the reference square is a parallelogram. A general quadrilateral requires a nonaffine map and is outside this affine definition; curved-cell approximation is not included here.

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

Oriented simplicial cell chains and boundary

Definition

Let $T$ be a matching simplicial mesh as in Matching affine simplicial meshes and shape regularity. Set $C_{- 1} (T; Z) = 0$ . For $0 \leq k \leq d$ , let $C_{k} (T; Z)$ be the free abelian group generated by ordered $k$ -faces $[v_{0}, \dots, v_{k}]$ , with the orientation relation $[v_{π (0)}, \dots, v_{π (k)}] = sgn (π) [v_{0}, \dots, v_{k}]$ for every permutation $π$ . Define $\partial_{0} : C_{0} \to C_{- 1}$ to be the zero map. For $1 \leq k \leq d$ , define the simplicial boundary homomorphism $\partial_{k} : C_{k} \to C_{k - 1}$ on a generator by $\partial_{k} [v_{0}, \dots, v_{k}] : = \sum_{j = 0}^{k} (- 1)^{j} [v_{0}, \dots, {\hat{v}}_{j}, \dots, v_{k}] .$ This is the oriented incidence convention; no geometric integration is part of the definition. The identity $\partial_{k - 1} \partial_{k} = 0$ is proved directly from this alternating formula in the following lemma.

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

The simplicial boundary of a boundary is zero

Statement

For every $1 \leq k \leq d$ , the boundary maps of Oriented simplicial cell chains and boundary satisfy $\partial_{k - 1} \circ \partial_{k} = 0.$

Facts & Assumptions

Given: An oriented simplicial mesh and its alternating boundary maps.

[F1]

An oriented generator has boundary $\partial_{k} [v_{0}, \dots, v_{k}] = \sum_{i = 0}^{k} (- 1)^{i} [v_{0}, \dots, {\hat{v}}_{i}, \dots, v_{k}]$ , extended linearly. (Oriented simplicial cell chains and boundary)

Proof

1.1F1given

If $k = 1$ , then $\partial_{k - 1} = \partial_{0} = 0$ by the definition, so the double boundary vanishes. Now let $k \geq 2$ and fix an oriented generator $[v_{0}, \dots, v_{k}]$ . In $\partial_{k - 1} \partial_{k} [v_{0}, \dots, v_{k}]$ , a codimension-two face is obtained by deleting a unique pair of vertices with indices $i < j$ . Deleting $i$ first contributes coefficient $(- 1)^{i} (- 1)^{j - 1}$ because the original index $j$ shifts down by one; deleting $j$ first contributes $(- 1)^{j} (- 1)^{i}$ .

2.1step 1.1algebra∎

The sum of those two coefficients is $(- 1)^{i + j - 1} + (- 1)^{i + j} = 0$ . Every codimension-two face arises from exactly these two deletion orders, so every basis coefficient in the double boundary is zero. The double boundary therefore vanishes on each generator and, by linearity, on all of $C_{k}$ .

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

Broken and conforming polynomial spaces on a matching mesh

Definition

Let $T$ be a matching affine simplicial mesh of $Ω$ as defined in Matching affine simplicial meshes and shape regularity. Fix $k \in N_{0}$ , $K \in {R, C}$ , and an integer $q \geq 1$ . On each cell let $P_{k} (K; K^{q})$ denote the restrictions to $K$ of $K^{q}$ -valued polynomials of total degree at most $k$ . The broken polynomial space is the finite product $V_{h}^{b r} : = \prod_{K \in T} P_{k} (K; K^{q}),$ viewed as functions defined almost everywhere cell by cell. For an interior facet $F = K \cap K^{'}$ , the jump is the difference of the two polynomial traces, with either fixed ordering of $K, K^{'}$ ; its vanishing is independent of that ordering. The conforming polynomial space is $V_{h}^{c o n f} : = {v \in V_{h}^{b r} : v ∣_{F, K} = v ∣_{F, K^{'}} on every interior facet F} .$ The zero-boundary subspace consists of those $v \in V_{h}^{c o n f}$ whose polynomial trace is zero on every boundary facet. These are algebraic definitions. The Sobolev inclusions and the zero-boundary closure conclusion are proved separately in the corresponding local theorems on matching piecewise polynomials and zero-boundary values. If $T = \emptyset$ , then $Ω = \emptyset$ by the mesh definition; the empty product is the zero vector space consisting of the unique empty function, and the trace conditions are vacuous.

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Reference finite elements and unisolvent degrees of freedom

Definition

Let $d \geq 1$ , let $\hat{K} \subset R^{d}$ be a nonempty bounded open reference cell, let $q \geq 1$ be an integer, and let $K \in {R, C}$ . A reference finite element is a triple $(\hat{K}, \hat{P}, \hat{Σ})$ where $\hat{P}$ is an $N$ -dimensional subspace of $[C (\overline{\hat{K}}; K)]^{q}$ for $N \in N_{0}$ and $\hat{Σ} = ({\hat{σ}}_{1}, \dots, {\hat{σ}}_{N})$ is a list of linear functionals on $\hat{P}$ for which $J_{\hat{Σ}} : \hat{P} \to K^{N}, J_{\hat{Σ}} (p) : = ({\hat{σ}}_{1} (p), \dots, {\hat{σ}}_{N} (p)),$ is bijective. The functionals are the degrees of freedom and the triple is unisolvent. Its shape functions are the unique dual basis ${\hat{ϕ}}_{1}, \dots, {\hat{ϕ}}_{N} \in \hat{P}$ satisfying ${\hat{σ}}_{i} ({\hat{ϕ}}_{j}) = δ_{i j}$ .

If $X$ is a normed vector space of functions on $\hat{K}$ containing $\hat{P}$ , fix for each $i$ a bounded linear extension of ${\hat{σ}}_{i}$ to $X$ , and denote this chosen extension by the same symbol. The reference interpolation operator associated with these chosen extensions is $\hat{I} : X \to \hat{P}, \hat{I} v : = \sum_{i = 1}^{N} {\hat{σ}}_{i} (v) {\hat{ϕ}}_{i} .$ The definition of a reference finite element does not assert that any particular list of functionals is unisolvent or bounded on a given Sobolev space; those are separate proof obligations. For $N = 0$ , $\hat{P} = {0}$ , $K^{0} = {0}$ , the empty list gives the unique bijection between these zero spaces, and the shape-function list is empty. The corresponding interpolation formula is the empty sum, hence the zero map into ${0}$ .

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Unisolvent degrees of freedom define a unique reference interpolant

Statement

Let $(\hat{K}, \hat{P}, \hat{Σ})$ be a reference finite element with $N = \dim \hat{P}$ , dual shape functions ${\hat{ϕ}}_{i}$ , and bounded extensions of its degrees of freedom to a normed function space $X$ containing $\hat{P}$ . Then the operator $\hat{I} v : = \sum_{i = 1}^{N} {\hat{σ}}_{i} (v) {\hat{ϕ}}_{i}$ is the unique linear map $X \to \hat{P}$ satisfying ${\hat{σ}}_{i} (\hat{I} v) = {\hat{σ}}_{i} (v)$ for every $i$ , and $\hat{I} p = p$ for every $p \in \hat{P}$ .

Facts & Assumptions

Given: The reference finite-element convention of Reference finite elements and unisolvent degrees of freedom and an input $v \in X$ on which every degree of freedom is defined.

[F1]

The degree-of-freedom map $J_{\hat{Σ}} : \hat{P} \to K^{N}$ is bijective, and the shape functions satisfy ${\hat{σ}}_{i} ({\hat{ϕ}}_{j}) = δ_{i j}$ . (Reference finite elements and unisolvent degrees of freedom)

Proof

1.1F1given

If $N = 0$ , then $\hat{P} = {0}$ and the empty sum defines the unique linear map $X \to \hat{P}$ ; all degree-of-freedom identities are vacuous and reproduction holds for the sole element $p = 0$ . For $N \geq 1$ , define $\hat{I} v = \sum_{i = 1}^{N} {\hat{σ}}_{i} (v) {\hat{ϕ}}_{i}$ . The sum is finite and each coefficient is linear in $v$ , so $\hat{I}$ is a well-defined linear map into $\hat{P}$ . Applying the $j$ th degree of freedom gives ${\hat{σ}}_{j} (\hat{I} v) = \sum_{i} {\hat{σ}}_{i} (v) {\hat{σ}}_{j} ({\hat{ϕ}}_{i}) = {\hat{σ}}_{j} (v)$ by [F1].

2.1F1step 1.1∎

If $p \in \hat{P}$ , then both $p$ and $\hat{I} p$ have the same degree-of-freedom vector, by step 1.1; injectivity of $J_{\hat{Σ}}$ therefore gives $\hat{I} p = p$ . If another $J : X \to \hat{P}$ has the same degree-of-freedom values, then $J_{\hat{Σ}} (J v - \hat{I} v) = 0$ , and injectivity again gives $J v = \hat{I} v$ . This proves uniqueness.

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Polynomial derivative and face norms are controlled on a reference simplex

Statement

Assume Countable Choice. Let $K \in {R, C}$ , ${\hat{S}}_{d}$ be the reference simplex, let $\hat{P}$ be a finite dimensional $K$ -space of polynomial functions on it, let $\hat{F}$ be any facet, and let integer $m \geq 0$ . For $1 \leq p, q \leq \infty$ there is a finite constant $C$ such that $∣ v ∣_{W^{m, p} ({\hat{S}}_{d})} + ∥ v ∥_{L^{p} (\hat{F})} \leq C ∥ v ∥_{L^{q} ({\hat{S}}_{d})} (v \in \hat{P}) .$ For every pair of integers $0 \leq m \leq ℓ$ , there is also a finite constant $C$ such that $∣ v ∣_{W^{ℓ, p} ({\hat{S}}_{d})} \leq C ∣ v ∣_{W^{m, q} ({\hat{S}}_{d})} (v \in \hat{P}) .$ For finitely many facets, one constant works for all of them. In particular, every linear degree-of-freedom functional on $\hat{P}$ is bounded with respect to the $L^{q} ({\hat{S}}_{d})$ norm.

Facts & Assumptions

Given: Countable Choice; the reference simplex has nonempty interior and finite positive volume; $\hat{P}$ has a finite ordered basis; Sobolev norms use the finite-derivative convention of Integer-order Sobolev spaces and their norms, and degrees of freedom use Reference finite elements and unisolvent degrees of freedom.

[F1]

If a finite-dimensional real vector space $V$ has positive dimension, choose an ordered basis and pull both norms back to $R^{\dim V}$ . The theorem on $R^{n}$ makes those pullbacks equivalent, and transporting the inequalities gives equivalence on $V$ ; the zero-dimensional case is immediate. (For $n \geq 1$ all norms on $R^{n}$ are equivalent, Equivalent norms, and the dictionary with equivalent metrics)

[F2]

On every finite-dimensional complex normed space, any two complex norms are equivalent. (All norms on a finite-dimensional complex normed space are equivalent, Equivalent complex norms)

[F3]

A degree of freedom is a linear functional on the finite-dimensional polynomial space. (Reference finite elements and unisolvent degrees of freedom)

[F4]

The Sobolev-space norm convention assumes Countable Choice. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F5]

The canonical open simplex and its facets are defined by Matching affine simplicial meshes and shape regularity; in particular the cell has nonempty interior and positive finite volume, and each facet is a bounded nondegenerate simplex of positive finite surface measure.

Proof

1.1F5givenalgebra

The cell expression $N_{0} (v) : = ∥ v ∥_{L^{q} ({\hat{S}}_{d})}$ is a norm on $\hat{P}$ . If it is zero, then for finite $q$ the continuous polynomial vanishes almost everywhere, and for $q = \infty$ it vanishes essentially everywhere. In either case, if it were nonzero at an interior point, continuity would make it bounded away from zero on an interior ball of positive measure, a contradiction. It therefore vanishes on the open simplex and hence is the zero polynomial, since its finite Taylor expansion at any interior point is exact.

2.1F1F2F3F4F5step 1.1algebra

If $\hat{P} = {0}$ , all displayed estimates and functional bounds hold immediately, so assume $\dim \hat{P} > 0$ . The expression $N_{1} (v) : = ∥ v ∥_{L^{p} ({\hat{S}}_{d})} + ∣ v ∣_{W^{m, p} ({\hat{S}}_{d})} + ∥ v ∥_{L^{p} (\hat{F})}$ is a norm: it is nonnegative, homogeneous and subadditive, and $N_{1} (v) = 0$ forces the cell $L^{p}$ norm to vanish, so the preceding argument gives $v = 0$ . For real coefficients apply [F1], and for complex coefficients apply [F2], to $N_{0}$ and $N_{1}$ ; norm equivalence gives $N_{1} (v) \leq C N_{0} (v)$ . Taking the maximum over the finitely many facets proves their uniform estimate. By [F3], any degree-of-freedom functional is linear on $\hat{P}$ ; it is bounded for $N_{1}$ by finite-dimensional norm equivalence and hence by $N_{0}$ .

3.1F1F2step 2.1algebra∎

If $\hat{P} = {0}$ , the second estimate is immediate. If $m = 0$ , the second estimate is the derivative part of step 2.1 with its order there set to $ℓ$ . If $m \geq 1$ , let $P_{m - 1}$ be the polynomials of degree at most $m - 1$ . The seminorm $∣ v ∣_{W^{m, q}}$ vanishes exactly on $\hat{P} \cap P_{m - 1}$ : every monomial of degree at least $m$ has a nonzero derivative of order $m$ , and every derivative of order $m$ kills lower-degree monomials. It is therefore a norm on the finite- dimensional quotient $\hat{P} / (\hat{P} \cap P_{m - 1})$ . If this quotient is zero, then the target seminorm vanishes on all of $\hat{P}$ because $ℓ \geq m$ , so the estimate is immediate. Otherwise add the target seminorm $∣ v ∣_{W^{ℓ, p}}$ to the quotient norm; it is well-defined because it also vanishes on $\hat{P} \cap P_{m - 1}$ . The sum is another norm. Applying [F1] or [F2] to these two norms bounds the target seminorm by the order- $m$ seminorm. This proves the second estimate.

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

Affine pullback of weak derivatives and Sobolev seminorms

Statement

Assume Countable Choice. Let $F (\hat{x}) = A \hat{x} + b$ be an affine bijection from a bounded open set $\hat{K} \subset R^{d}$ to $K$ , where $A$ is invertible, and let $1 \leq p \leq \infty$ . For every integer $r \geq 0$ and scalar $v \in W^{r, p} (K)$ , the pullback $\hat{v} : = v \circ F$ belongs to $W^{r, p} (\hat{K})$ . For $r \geq 1$ its first weak derivatives satisfy $D_{{\hat{x}}_{j}} \hat{v} = \sum_{i = 1}^{d} A_{i j} (D_{x_{i}} v) \circ F,$ and higher derivatives are obtained by applying this constant-matrix rule repeatedly. In particular, for each $0 \leq m \leq r$ , $∣ \hat{v} ∣_{W^{m, p} (\hat{K})} \leq C (d, m) ∥ A ∥^{m} ∣ \det A ∣^{- 1 / p} ∣ v ∣_{W^{m, p} (K)},$ where $∣ \det A ∣^{- 1 / p} : = 1$ for $p = \infty$ . The reverse estimate holds with $A^{- 1}$ and $∣ \det A ∣^{1 / p}$ in place of $A$ and $∣ \det A ∣^{- 1 / p}$ .

Facts & Assumptions

Given: Countable Choice; the affine bijection $F$ ; an exponent $1 \leq p \leq \infty$ ; and $v \in W^{r, p} (K)$ .

[F1]

A weak derivative is characterized by $\int_{K} v D^{α} ϕ = (- 1)^{∣ α ∣} \int_{K} (D^{α} v) ϕ$ for all compactly supported smooth tests. (Weak derivative of a locally integrable function, Integer-order Sobolev spaces and their norms)

[F2]

For a C1 diffeomorphism, nonnegative Lebesgue-measurable integrands obey the change-of-variables formula with $∣ \det D F ∣$ ; Countable Choice is an explicit hypothesis. (A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions)

[F3]

$A$ is constant, so the classical chain rule for a test function is $D_{{\hat{x}}_{j}} (ϕ \circ F) = \sum_{i} A_{i j} (D_{x_{i}} ϕ) \circ F$ . (The chain rule for differentials of smooth maps)

[F4]

For finite $p$ , $∥ g \circ F ∥_{L^{p} (\hat{K})} = ∣ \det A ∣^{- 1 / p} ∥ g ∥_{L^{p} (K)}$ by [F2]; for $p = \infty$ the essential-supremum norm is preserved by a C1 bijection and its inverse, which map null sets to null sets. (A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions)

[F5]

The cited change-of-variables theorem assumes Countable Choice. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

Proof

1.1F1F2F3F4F5given

If $\hat{K} = \emptyset$ , then $K = \emptyset$ and each Sobolev space on these measure-zero domains has only the zero class; every displayed derivative identity and estimate is therefore zero equals zero. Otherwise, if $r = 0$ , the only derivative is order zero and [F4] gives $∥ v \circ F ∥_{L^{p} (\hat{K})} = ∣ \det A ∣^{- 1 / p} ∥ v ∥_{L^{p} (K)}$ , with the stated convention at $p = \infty$ . Thus the membership and pullback estimate hold in this case. For $r \geq 1$ , fix $j$ and $ψ \in C_{c}^{\infty} (\hat{K})$ . Set $ϕ : = ψ \circ F^{- 1} \in C_{c}^{\infty} (K)$ . By [F1], $v$ and its weak first derivatives have locally integrable representatives, so $v D_{x_{i}} ϕ$ and $(D_{x_{i}} v) ϕ$ are integrable: the test factors are bounded and compactly supported. For an integrable real product $g$ on $K$ , apply [F2] separately to its nonnegative parts $g^{+}$ and $g^{-}$ ; both integrals are finite, and the constant $∣ \det A ∣ > 0$ makes the corresponding pullback integrals finite. Subtracting their two formulas gives $\int_{\hat{K}} g \circ F = ∣ \det A ∣^{- 1} \int_{K} g$ . For a complex product, apply this argument to its real and imaginary parts and combine their integrals componentwise; the weak test pairing remains bilinear, without conjugation. Changing variables by [F2] and using [F3] gives $\int_{\hat{K}} (v \circ F) D_{{\hat{x}}_{j}} ψ d \hat{x} = ∣ \det A ∣^{- 1} \sum_{i} A_{i j} \int_{K} v D_{x_{i}} ϕ d x = - ∣ \det A ∣^{- 1} \sum_{i} A_{i j} \int_{K} (D_{x_{i}} v) ϕ d x = - \sum_{i} A_{i j} \int_{\hat{K}} ((D_{x_{i}} v) \circ F) ψ d \hat{x} .$ Thus the displayed first-derivative formula holds weakly; its right side is in $L^{p}$ by [F4].

2.1F3F4F5step 1.1algebra∎

For $r \geq 1$ , since $A$ is constant, differentiating the first formula weakly again applies the same rule to each $D_{x_{i}} v$ ; induction on $m$ gives every derivative of order $m \leq r$ as a sum of at most $d^{m}$ terms, each with a product of $m$ entries of $A$ multiplying a pullback of an order- $m$ derivative of $v$ . Taking component $L^{p}$ norms and using [F4] yields the stated seminorm bound, with a finite combinatorial constant depending on $d, m$ . Applying the proved estimate to $F^{- 1} (x) = A^{- 1} (x - b)$ gives the reverse bound. Together with step 1.1 this includes $r = 0$ and proves both directions.

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

Shape regularity controls affine simplex geometry

Statement

Assume Countable Choice. Let $F_{K} (\hat{x}) = b_{K} + A_{K} \hat{x}$ map the fixed reference simplex ${\hat{S}}_{d}$ bijectively to a cell $K$ of a shape-regular family, and let $h_{K}$ be its diameter and $ρ_{K}$ its inradius. There are constants $c, C > 0$ , depending only on $d$ , the reference simplex, and the shape regularity bound, such that $c h_{K} \leq s_{\min} (A_{K}) \leq ∥ A_{K} ∥ \leq C h_{K}, c h_{K}^{d} \leq \det A_{K} \leq C h_{K}^{d} .$ For each facet $F$ , its affine surface Jacobian is comparable to $h_{K}^{d - 1}$ , and its altitude from the opposite vertex is comparable to $h_{K}$ . Thus Sobolev scaling constants can be expressed using $h_{K}$ alone, with constants uniform over the family.

Facts & Assumptions

Given: A nondegenerate affine simplex map, a fixed reference simplex, and the shape regularity bound $h_{K} / ρ_{K} \leq σ$ .

[F1]

The reference simplex is fixed, with diameter $D_{w i d e h a t S} < \infty$ and inradius $r_{w i d e h a t S} > 0$ ; $h_{K}$ is the diameter and $ρ_{K}$ the inradius of $K$ . (Matching affine simplicial meshes and shape regularity)

[F2]

The oriented area element of an affine facet is $n d S = \det (A_{K}) A_{K}^{- T} \hat{n} d \hat{S}$ , so its surface Jacobian is $J_{F} = \det (A_{K}) ∣ A_{K}^{- T} \hat{n} ∣$ when $\det A_{K} > 0$ . Affine Sobolev seminorms transform with the corresponding powers of $A_{K}$ and $A_{K}^{- 1}$ . (Gauss–Green formula on an affine simplex, Affine pullback of weak derivatives and Sobolev seminorms)

[F3]

The two affine change-of-variables suppliers used in [F2] assume Countable Choice. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

Proof

1.1F1givenalgebra

Use a singular-value decomposition $A_{K} = U Σ V^{T}$ . If $v_{\max}$ is a unit right singular vector for $s_{\max} = ∥ A_{K} ∥$ , the reference inscribed ball contains two points separated by $2 r_{\hat{S}} v_{\max}$ ; their images are separated by $2 r_{\hat{S}} s_{\max}$ . Thus $s_{\max} \leq h_{K} / (2 r_{\hat{S}})$ . For a unit right singular vector $v_{\min}$ and corresponding left vector $u_{\min}$ , the width of $K$ in direction $u_{\min}$ equals $s_{\min}$ times the width of the reference simplex in direction $v_{\min}$ , which is at most $s_{\min} D_{\hat{S}}$ . Every ball of radius $t < ρ_{K}$ contained in $K$ has width $2 t$ in every direction, so letting $t ↑ ρ_{K}$ gives $2 ρ_{K} \leq s_{\min} D_{\hat{S}}$ . Since $h_{K} / ρ_{K} \leq σ$ , $s_{\min} \geq 2 h_{K} / (σ D_{\hat{S}})$ .

2.1step 1.1givenalgebra

Every singular value lies between $c h_{K}$ and $C h_{K}$ by step 1.1. Their product is $∣ \det A_{K} ∣ = \det A_{K}$ because the reference map is orientation preserving. Hence $c^{'} h_{K}^{d} \leq \det A_{K} \leq C^{'} h_{K}^{d}$ .

3.1F2F3step 1.1step 2.1algebra∎

For a reference facet normal $\hat{n}$ , [F2] gives its physical surface Jacobian $J_{F} = \det (A_{K}) ∣ A_{K}^{- T} \hat{n} ∣$ . The singular values of $A_{K}$ from step 1.1 are between $c h_{K}$ and $C h_{K}$ , so $∣ A_{K}^{- T} \hat{n} ∣$ is between $c h_{K}^{- 1}$ and $C h_{K}^{- 1}$ . Together with the determinant bounds this gives $c h_{K}^{d - 1} \leq J_{F} \leq C h_{K}^{d - 1}$ . The simplex pyramid identity $∣ K ∣ = ∣ F ∣ H_{F} / d$ gives $H_{F} = d ∣ K ∣ / ∣ F ∣$ , so step 2.1 and these facet bounds show $c h_{K} \leq H_{F} \leq C h_{K}$ . This proves all claimed geometric scalings.

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

Gauss–Green formula on an affine simplex

Statement

Assume Countable Choice. Let $K$ be a nondegenerate affine $d$ -simplex and let $G \in C^{1} (\overline{K}; R^{d})$ . If $n_{K, F}$ is the outward unit normal on each facet $F$ , then $\int_{K} div G d x = \sum_{F \subset \partial K} \int_{F} G \cdot n_{K, F} d S .$ For an orientation-preserving affine map $F_{K} (\hat{x}) = b + A \hat{x}$ , the oriented area element transforms as $n_{K, F} d S = \det (A) A^{- T} {\hat{n}}_{\hat{F}} d \hat{S} .$

Facts & Assumptions

Given: Countable Choice, a nondegenerate affine simplex $K$ , its fixed reference simplex, and a continuously differentiable vector field on $\overline{K}$ .

[F1]

The reference open simplex is ${{\hat{x}}_{i} > 0, \sum_{i} {\hat{x}}_{i} < 1}$ ; every physical cell is its affine image. (Matching affine simplicial meshes and shape regularity)

[F2]

For affine maps, weak derivatives transform by the constant Jacobian chain rule, and volume measure transforms by $∣ \det A ∣$ . (Affine pullback of weak derivatives and Sobolev seminorms)

[F3]

The one-dimensional fundamental theorem of calculus gives $\int_{a}^{b} f^{'} (t) d t = f (b) - f (a)$ for a continuously differentiable vector-valued function. (If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz)

[F4]

Fubini's theorem permits iterated integration of the continuous bounded functions on the bounded simplex used below. (Tonelli and Fubini for the completed product, with only almost-everywhere section measurability)

[F5]

A C1 diffeomorphism obeys Lebesgue change of variables with the absolute determinant. For the bounded integrands here, apply the nonnegative formula separately to positive and negative parts. (A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions)

[F6]

The cited change-of-variables theorem assumes Countable Choice. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

Proof

1.1F1F3F4givenalgebra

First take ${\hat{S}}_{d}$ . Fix $i$ and write $y = ({\hat{x}}_{j})_{j \neq i}$ , where $y_{j} > 0$ and $\sum_{j \neq i} y_{j} < 1$ . The $i$ th coordinate ranges from $0$ to $g (y) : = 1 - \sum_{j \neq i} y_{j}$ . By [F3], $\int_{0}^{g (y)} \partial_{i} G_{i} (y, t) d t = G_{i} (y, g (y)) - G_{i} (y, 0) .$ Integrating in $y$ and using [F4] gives the volume integral of $\partial_{i} G_{i}$ as the difference of these two graph integrals. The lower graph is the coordinate facet with outward normal $- e_{i}$ and area factor $1$ , hence its flux contribution is $- G_{i} (y, 0)$ . The upper facet is the graph ${\hat{x}}_{i} = 1 - \sum_{j \neq i} {\hat{x}}_{j}$ : its tangent Gram matrix is $I_{d - 1} + 11^{T}$ , whose eigenvalues are $1$ (multiplicity $d - 2$ ) and $d$ , so its area factor is $\sqrt{d}$ . Its outward unit normal is $d^{- 1 / 2} (1, \dots, 1)$ , so the $i$ th component of normal times area factor is $1$ . Thus the upper graph integral is exactly its $i$ th flux contribution. Other coordinate facets have zero $i$ th normal component. Summing over $i$ accounts for every component of the boundary flux and proves the formula on ${\hat{S}}_{d}$ .

1.2F1givenalgebra

Let ${\hat{τ}}_{1}, \dots, {\hat{τ}}_{d - 1}$ be an oriented tangent basis for a reference facet, with orientation chosen so its oriented area vector is $\hat{n} d \hat{S}$ . By the determinant definition of oriented area vector, for every $w \in R^{d}$ , $w \cdot (n_{K, F} d S) = \det (A {\hat{τ}}_{1}, \dots, A {\hat{τ}}_{d - 1}, w) = \det (A) \det ({\hat{τ}}_{1}, \dots, {\hat{τ}}_{d - 1}, A^{- 1} w) = w \cdot (\det (A) A^{- T} \hat{n} d \hat{S}) .$ Equality against every $w$ gives the cofactor area-vector formula. Because $\det A > 0$ , $A^{- T} \hat{n}$ points to the exterior side corresponding to $\hat{n}$ , so the formula has the outward orientation.

2.1F2F5F6step 1.1step 1.2algebra∎

Define $\hat{G} (\hat{x}) : = \det (A) A^{- 1} G (b + A \hat{x})$ . The chain rule gives ${div}_{\hat{x}} \hat{G} = \det (A) \sum_{j, i, l} (A^{- 1})_{j i} (\partial_{x_{l}} G_{i}) (F (\hat{x})) A_{l j} = \det (A) ({div}_{x} G) (F (\hat{x})),$ since $\sum_{j} A_{l j} (A^{- 1})_{j i} = δ_{l i}$ . Applying the reference identity of step 1.1 to $\hat{G}$ , and using [F5] to change variables in its bounded volume integrand, gives the physical volume integral. On each facet, step 1.2 gives $\hat{G} \cdot \hat{n} d \hat{S} = G (F (\hat{x})) \cdot \det (A) A^{- T} \hat{n} d \hat{S} = G \cdot n_{K, F} d S .$ Summing the facet identities proves the stated Gauss–Green formula on $K$ .

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Smooth functions are dense in finite-order Sobolev spaces on a reference simplex

Statement

Assume Countable Choice. Let $d \geq 2$ , $k \in N_{0}$ , $1 \leq p < \infty$ , and let ${\hat{S}}_{d}$ be the reference simplex. Then restrictions of functions smooth on a neighbourhood of $\overline{{\hat{S}}_{d}}$ are dense in $W^{k, p} ({\hat{S}}_{d})$ in its Sobolev norm. No density assertion for $p = \infty$ is made.

Facts & Assumptions

Given: The affine reference simplex and Sobolev conventions of Matching affine simplicial meshes and shape regularity and Integer-order Sobolev spaces and their norms, with finite $p$ and Countable Choice.

[F1]

For an invertible affine map $F (x) = A x + b$ , weak derivatives obey $D_{j} (u \circ F) = \sum_{i} A_{i j} (D_{i} u) \circ F$ ; higher derivatives follow by iterating this constant-matrix rule, with the corresponding determinant norm factors. For a homothety with $A = t I$ , this gives $D^{α} (u \circ F) = t^{∣ α ∣} (D^{α} u) \circ F$ . (Affine pullback of weak derivatives and Sobolev seminorms)

[F2]

For finite $p$ , $C_{c}^{\infty} (R^{d})$ is dense in $W^{k, p} (R^{d})$ ; in particular its order-zero case gives density in $L^{p} (R^{d})$ . (Compactly supported smooth functions are dense in W^{k,p}(R^n))

[F3]

Smooth cutoffs equal to one near a compact subset of an open set exist; multiplication by a smooth function with bounded derivatives preserves $W^{k, p}$ ; and a compactly supported $W^{k, p}$ function on an open set extends by zero to $R^{d}$ . (Test function cutoffs and euclidean localization, Weak Leibniz rule with a smooth factor, Compactly supported Sobolev functions extend by zero in every integer order)

[F4]

Countable Choice is the explicit assumption in the cited Sobolev, weak-derivative and density interfaces. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

Proof

1.1F1givenalgebra

Let $a$ be the barycenter of ${\hat{S}}_{d}$ and, for $0 < t < 1$ , set $F_{t} (x) = a + t (x - a)$ . The set $S_{t} : = F_{t}^{- 1} ({\hat{S}}_{d})$ is open and contains $\overline{{\hat{S}}_{d}}$ : each point of the closure is carried to a strict convex combination of that point and the interior point $a$ , which lies in the open simplex. Thus $u_{t} : = u \circ F_{t}$ is defined on $S_{t}$ for $u \in W^{k, p} ({\hat{S}}_{d})$ , and [F1] gives $D^{α} u_{t} = t^{∣ α ∣} (D^{α} u) \circ F_{t}$ .

2.1F1F2step 1.1given

For every $g \in L^{p} ({\hat{S}}_{d})$ , $g \circ F_{t} \to g$ in $L^{p} ({\hat{S}}_{d})$ as $t ↑ 1$ . To see this, extend $g$ by zero to $R^{d}$ and approximate that extension in $L^{p}$ by $ϕ \in C_{c}^{\infty} (R^{d})$ using [F2]. For $t \geq 1 / 2$ , change of variables bounds the pullback error by $t^{- d / p} ∥ g - ϕ ∥_{L^{p} (R^{d})} \leq 2^{d / p} ∥ g - ϕ ∥_{p}$ ; for fixed $ϕ$ , uniform continuity on the compact closure gives $ϕ \circ F_{t} \to ϕ$ uniformly there. First choosing $ϕ$ with arbitrarily small error and then taking $t ↑ 1$ proves the claim. Applying it to each of the finitely many $D^{α} u$ and using [F1] shows $u_{t} ∣_{{\hat{S}}_{d}} \to u$ in $W^{k, p} ({\hat{S}}_{d})$ .

2.2F2F3step 1.1

Fix $t < 1$ . Since $\overline{{\hat{S}}_{d}}$ is compactly contained in $S_{t}$ , choose $η \in C_{c}^{\infty} (S_{t})$ equal to one near that closure by [F3]. Then $η u_{t} \in W^{k, p} (S_{t})$ has compact support in $S_{t}$ , so its zero extension $U_{t}$ belongs to $W^{k, p} (R^{d})$ by [F3]. By [F2], choose $ϕ_{j} \in C_{c}^{\infty} (R^{d})$ converging to $U_{t}$ in that norm. Since $U_{t} = u_{t}$ on a neighbourhood of the closed simplex, the restrictions $ϕ_{j} ∣_{{\hat{S}}_{d}}$ converge to $u_{t} ∣_{{\hat{S}}_{d}}$ in $W^{k, p} ({\hat{S}}_{d})$ .

3.1F4step 2.1step 2.2algebra∎

Given $ε > 0$ , choose $t$ close enough to one that $∥ u_{t} - u ∥_{W^{k, p} ({\hat{S}}_{d})} < ε / 2$ by step 2.1, and then choose $j$ in step 2.2 with $∥ ϕ_{j} - u_{t} ∥_{W^{k, p} ({\hat{S}}_{d})} < ε / 2$ . The triangle inequality gives $∥ ϕ_{j} - u ∥_{W^{k, p} ({\hat{S}}_{d})} < ε$ . The choices are finite for each $u, ε$ ; Countable Choice is only the declared background of [F4], and no $p = \infty$ density is used.

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A reference simplex is an integer-order Sobolev extension domain

Statement

Assume the Axiom of Choice. Let $d \geq 2$ . For every integer $k \geq 0$ and $1 \leq p \leq \infty$ , there is a bounded linear operator $E_{k, p} : W^{k, p} ({\hat{S}}_{d}) ⟶ W^{k, p} (R^{d})$ whose restriction to ${\hat{S}}_{d}$ is the input class. Consequently ${\hat{S}}_{d}$ is a $W^{k, p}$ -extension domain in the sense of Sobolev extension domains and extension operators. The operator may be chosen with support in one fixed bounded set.

Facts & Assumptions

Given: $d \geq 2$ , the reference simplex, $k \in N_{0}$ , $1 \leq p \leq \infty$ , and the Axiom of Choice.

[F1]

Affine pullback and pushforward preserve $W^{k, p}$ , with weak derivatives and norm bounds controlled by the fixed affine matrix. (Affine pullback of weak derivatives and Sobolev seminorms)

[F2]

On a half-space, for every integer $k$ and $p$ there is an explicit bounded extension built from finitely many reflected values; its coefficients satisfy $\sum_{j} a_{j} (- j)^{r} = 1$ for $0 \leq r < k$ , so normal derivatives through order $k - 1$ match at the reflecting hyperplane. Its proof verifies the weak derivative identity by ACL sections, integration by parts and Fubini, and gives a bound depending only on $k, p, d$ . (Integer-order Sobolev extension from a half-space)

[F3]

Smooth cutoffs with compact support in a prescribed chart exist; the weak Leibniz rule bounds localized Sobolev norms; and a function supported away from the artificial edge of an open chart can be extended by zero across that edge by the weak test identity. (Test function cutoffs and euclidean localization, Weak Leibniz rule with a smooth factor, Compactly supported Sobolev functions extend by zero in every integer order)

[F4]

The closed reference simplex is compact, and a finite subcover can be selected from any open cover of it. (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line)

[F5]

The Axiom of Choice implies each finite sequence of affine charts, cutoffs and local extension maps needed below can be selected; the local half-space extension theorem itself assumes this axiom. (The Axiom of Choice)

[F6]

A $W^{k, p}$ -extension domain admits a bounded linear operator to $W^{k, p} (R^{d})$ whose restriction is the input almost-everywhere class. (Sobolev extension domains and extension operators)

Proof

1.1givenalgebra

Let $z \in \overline{{\hat{S}}_{d}}$ . Define the barycentric affine coordinates by $λ_{0} (x) = 1 - \sum_{i = 1}^{d} x_{i}$ and $λ_{i} (x) = x_{i}$ for $1 \leq i \leq d$ ; they sum to one and are nonnegative exactly on the closed simplex. The active set $I (z) = {i : λ_{i} (z) = 0}$ has at most $d$ elements, and the gradients of the corresponding coordinates are linearly independent. Complete these affine coordinates to an invertible affine coordinate system $y = (λ_{i})_{i \in I (z)}, y_{m + 1}, \dots, y_{d}$ , where $m = ∣ I (z) ∣$ . Because every inactive barycentric coordinate is strictly positive at $z$ , a sufficiently small neighbourhood $U_{z}$ meets the simplex exactly in the orthant chart $y_{1} > 0, \dots, y_{m} > 0$ ; for $m = 0$ it is an interior chart.

2.1F3F4step 1.1

The charts $U_{z}$ cover the compact closure, so [F4] gives a finite subcover. Shrink them to open sets $V_{j}$ still covering the closure, with ${\overline{V}}_{j} \subset U_{j}$ . Use [F3] to choose smooth cutoffs $θ_{j}$ supported compactly in $U_{j}$ and equal to one on $V_{j}$ . Their sum $S$ is positive on a neighbourhood $N$ of the closed simplex. Choose a smooth cutoff $χ$ supported compactly in $N$ and equal to one near the closed simplex, and set $η_{j} = χ θ_{j} / S$ on $N$ , zero outside $N$ . Then each $η_{j}$ is smooth and supported in its chart, and their sum is one near the simplex. Each localized input $η_{j} u$ has norm bounded by a fixed multiple of $∥ u ∥_{W^{k, p} ({\hat{S}}_{d})}$ by the product rule in [F3].

3.1F3step 2.1

In an interior chart, the localized function is zero near the chart edge. Its zero extension to $R^{d}$ has the same weak derivatives: for any test function, multiply that test by a cutoff equal to one on the localized support, apply the weak identity on the chart, and note that all derivatives of the cutoff product agree on that support. This is the zero-extension part of [F3]. In a boundary chart with $m$ active faces, the localized function is a $W^{k, p}$ function on the orthant $O_{m} = (0, \infty)^{m} \times R^{d - m}$ and vanishes near the artificial edge of the chart; extend by zero across only that artificial edge using the same test-function argument.

4.1F1F2step 3.1

Extend from $O_{m}$ to $R^{d}$ one coordinate at a time. At each stage, apply the reflected half-space formula in [F2] to the coordinate being extended, with the other $m - 1$ positive coordinates as tangential variables. For clarity, the coordinate formula on $t < 0$ is $\sum_{j = 1}^{k} a_{j} w (x^{'}, - j t)$ , with $\sum_{j} a_{j} (- j)^{r} = 1$ for $0 \leq r < k$ ; for $k = 0$ use even reflection. For a test compactly supported in $O_{m - 1} \times R$ , apply the half-space proof's normal ACL trace and integration-by-parts identity on each compact tangential subdomain and then exhaust the test support by finitely many such subdomains. Tangential weak derivatives pass through the finite reflection sum by the weak test identity, and the displayed moment equations cancel every normal interface term. The change of variable $s = - j t$ bounds each reflected derivative in $L^{p}$ by the same finite coefficient sum as in [F2] (and by the corresponding essential supremum sum at $p = \infty$ ). Thus this explicit operator is bounded on $W^{k, p} (O_{m})$ and extends one orthant coordinate. Repeating at most $m \leq d$ times gives the required extension. Pulling back through the affine chart changes the bound by a finite factor from [F1].

5.1F5F6step 2.1step 4.1∎

Apply these local bounded linear extensions to the finitely many localized pieces and add them. Their sum is bounded and linear, and on ${\hat{S}}_{d}$ its restriction is $\sum_{j} η_{j} u = u$ . There are finitely many charts, all have bounded coordinate images and the reflection operators add support only by fixed finite factors, so the sum is supported in a fixed bounded set. The definition of extension domain in [F6] gives the conclusion.

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Sobolev face traces and Gauss–Green on a reference simplex

Statement

Assume Countable Choice. Let $1 < p < \infty$ and $p^{'} = p / (p - 1)$ . For each facet $F$ of ${\hat{S}}_{d}$ , restriction of smooth functions extends uniquely to a bounded linear trace map $γ_{F} : W^{1, p} ({\hat{S}}_{d}; R) \to L^{p} (F; R) .$ For real-valued $u \in W^{1, p} ({\hat{S}}_{d})$ and $G \in W^{1, p^{'}} ({\hat{S}}_{d}; R^{d})$ , the facet traces satisfy $\int_{{\hat{S}}_{d}} u div G + \nabla u \cdot G d x = \sum_{F \subset \partial {\hat{S}}_{d}} \int_{F} γ_{F} u γ_{F} G \cdot {\hat{n}}_{F} d S .$ Here $γ_{F} G$ is the componentwise trace and ${\hat{n}}_{F}$ is the outward unit normal. The bounded C1-domain trace and Green theorems are not used for the simplex.

Facts & Assumptions

Given: Countable Choice, the reference simplex, $1 < p < \infty$ , and the Sobolev spaces specified in Integer-order Sobolev spaces and their norms.

[F1]

Smooth functions on a neighbourhood of the closed simplex are dense in $W^{1, q} ({\hat{S}}_{d})$ for every finite $q$ . (Smooth functions are dense in finite-order Sobolev spaces on a reference simplex)

[F2]

For $C^{1}$ vector fields on a nondegenerate affine simplex, the divergence integral equals the sum of outward facet fluxes. (Gauss–Green formula on an affine simplex)

[F3]

The spaces $L^{p} (F)$ are complete for finite $p$ , and Holder's inequality gives $∥ f g ∥_{L^{1} (F)} \leq ∥ f ∥_{L^{p} (F)} ∥ g ∥_{L^{p^{'}} (F)}$ . (Riesz-Fischer completeness of $L^{p}$ for $1 \leq p \leq \infty$ , Holder's inequality for integrals, including the endpoint cases)

[F4]

The smooth-density supplier [F1] assumes Countable Choice. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

Proof

1.1F2F3givenalgebra

Fix a facet $F$ opposite the vertex $z_{F}$ , let $H_{F} > 0$ be the altitude from $z_{F}$ to $F$ , and put $X (x) = x - z_{F}$ . On $F$ , $X \cdot {\hat{n}}_{F} = H_{F}$ ; on every other facet, which contains $z_{F}$ , $X$ is tangent, so its normal component is zero. For a smooth scalar $u$ , apply [F2] to $∣ u ∣^{p} X$ . Since $p > 1$ , $∣ u ∣^{p}$ is continuously differentiable, and $∣ X ∣ \leq diam ({\hat{S}}_{d})$ ; hence $H_{F} \int_{F} ∣ u ∣^{p} d S \leq d \int_{{\hat{S}}_{d}} ∣ u ∣^{p} d x + p diam ({\hat{S}}_{d}) \int_{{\hat{S}}_{d}} ∣ u ∣^{p - 1} ∣ \nabla u ∣ d x .$ Holder bounds the last integral by $∥ u ∥_{p}^{p - 1} ∥ \nabla u ∥_{p}$ . For $a = ∥ u ∥_{p}$ and $b = ∥ \nabla u ∥_{p}$ , if $b \leq a$ then $a^{p - 1} b \leq a^{p}$ , while if $a \leq b$ then $a^{p - 1} b \leq b^{p}$ . Thus the product is at most $a^{p} + b^{p}$ . Since $H_{F} > 0$ , the resulting inequality gives $∥ u ∥_{L^{p} (F)} \leq C_{F, p} ∥ u ∥_{W^{1, p} ({\hat{S}}_{d})}$ .

2.1F1F3step 1.1

Let $(u_{j})$ be a smooth sequence converging to $u$ in $W^{1, p} ({\hat{S}}_{d})$ , which exists by [F1]. Applying the estimate of step 1.1 to $u_{j} - u_{i}$ shows $(u_{j} ∣_{F})$ is Cauchy in the complete space $L^{p} (F)$ ; define $γ_{F} u$ to be its limit. The same estimate shows the limit is independent of the chosen smooth approximating sequence and $∥ γ_{F} u ∥_{L^{p} (F)} \leq C ∥ u ∥_{W^{1, p}}$ . Linearity follows by taking approximating sequences for sums and scalar multiples.

3.1F1F2F3F4step 2.1∎

For smooth $u, G$ , apply [F2] to the vector field $u G$ . Its divergence is $\nabla u \cdot G + u div G$ , so the displayed Green identity holds with ordinary restrictions. Choose smooth approximations $u_{j} \to u$ in $W^{1, p}$ and $G_{j} \to G$ componentwise in $W^{1, p^{'}}$ by [F1]. By step 2.1 their facet traces converge in $L^{p}$ and $L^{p^{'}}$ ; Holder [F3] makes the boundary products converge in $L^{1} (F)$ . The volume products converge in $L^{1}$ by Holder and convergence of both Sobolev norms. Passing to the limit in the smooth identity proves the formula for $u, G$ .

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Continuous matching piecewise polynomials belong to W1p

Statement

Assume Countable Choice. Let $T$ be a finite matching affine simplicial mesh of a bounded polyhedral open set $Ω$ , let $1 \leq p \leq \infty$ , and let $v$ be a real-valued scalar member of the conforming piecewise-polynomial space in Broken and conforming polynomial spaces on a matching mesh. Then $v \in W^{1, p} (Ω)$ . Its weak gradient agrees almost everywhere on each cell with the ordinary cellwise polynomial gradient.

Facts & Assumptions

Given: Countable Choice, a finite matching mesh, $v$ polynomial on each cell with equal polynomial traces across every interior facet, $1 \leq p \leq \infty$ , and $φ \in C_{c}^{\infty} (Ω)$ .

[F1]

The conforming condition is equality of the two polynomial traces on each whole interior facet. (Broken and conforming polynomial spaces on a matching mesh)

[F2]

A $C^{1}$ field on a simplex satisfies the cellwise Gauss–Green formula. (Gauss–Green formula on an affine simplex)

[F3]

The two outward normals on an interior matching facet are opposite, so its oriented contributions cancel when cell boundaries are summed. (Interior facet orientations cancel in a matching mesh)

[F4]

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

[F5]

Countable Choice is the explicit background assumption in the Sobolev-space definition. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

Proof

1.1F2given

If $T = \emptyset$ , then $Ω = \emptyset$ by the mesh definition; $v$ is the unique zero class in $W^{1, p} (Ω)$ and the cellwise-gradient assertion has no cells. Assume now $T \neq \emptyset$ . The mesh is finite and each cellwise polynomial and its ordinary first derivatives are bounded on the closed cell. Thus $v$ and the piecewise function $g_{i} = \partial_{i} (v ∣_{K})$ belong to $L^{p} (Ω)$ for every $1 \leq p \leq \infty$ . Fix a coordinate $i$ and a test function $φ \in C_{c}^{\infty} (Ω)$ . On each cell apply [F2] to $G = (v φ) e_{i}$ to obtain $\int_{K} v \partial_{i} φ d x = - \int_{K} g_{i} φ d x + \sum_{F \subset \partial K} \int_{F} v φ (n_{K, F})_{i} d S .$

2.1F1F3F4F5step 1.1∎

Sum these finitely many identities over $K$ . On an interior facet, [F1] gives a common trace for $v φ$ and [F3] gives opposite outward normals, so the two flux integrals cancel. On a boundary facet, $φ = 0$ because its support is compact in $Ω$ . Therefore $\int_{Ω} v \partial_{i} φ d x = - \int_{Ω} g_{i} φ d x .$ By [F4], $g_{i}$ is the weak $i$ th derivative. This holds for every coordinate $i$ ; all $g_{i}$ are in $L^{p}$ , so $v \in W^{1, p} (Ω)$ with the stated cellwise gradient.

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Zero-boundary conforming polynomials lie in W0 1p and satisfy Poincare

Statement

Assume Countable Choice. Let $Ω$ be a bounded polyhedral open set with a finite matching affine simplicial mesh, let $1 \leq p < \infty$ , and let $v$ be a real-valued conforming piecewise polynomial whose polynomial trace vanishes on every boundary facet. Then $v \in W_{0}^{1, p} (Ω), ∥ v ∥_{L^{p} (Ω)} \leq C_{Ω} ∥ \nabla v ∥_{L^{p} (Ω)} .$ Here $W_{0}^{1, p}$ is the norm closure of $C_{c}^{\infty} (Ω)$ ; no C1 trace characterization is used.

Facts & Assumptions

Given: Countable Choice, the finite matching mesh, the conforming and zero-boundary conditions, bounded $Ω$ , and finite $p$ .

[F1]

On each simplex, a $C^{1}$ vector field satisfies the cellwise Gauss–Green formula. (Gauss–Green formula on an affine simplex)

[F2]

The mesh is the interior of the union of its cells, so each boundary component of a local simplicial link reaches an exterior facet. Equal polynomial traces and opposite facet orientations cancel interior fluxes; the zero-boundary condition makes every exterior facet trace zero. (Matching affine simplicial meshes and shape regularity, Broken and conforming polynomial spaces on a matching mesh, Interior facet orientations cancel in a matching mesh)

[F3]

The zero-boundary space is precisely the norm closure of $C_{c}^{\infty} (Ω)$ in $W^{1, p} (Ω)$ . (Zero-boundary Sobolev space as a norm closure)

[F4]

If $Ω$ is bounded in one direction, every $w \in W_{0}^{1, p} (Ω)$ satisfies $∥ w ∥_{L^{p}} \leq C (p) (b - a) ∥ \nabla w ∥_{L^{p}}$ for a slab of width $b - a$ and $1 \leq p < \infty$ . (The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction)

[F5]

Compactly supported Sobolev functions extend by zero, and compactly supported $W^{1, p} (R^{d})$ functions can be approximated by smooth compactly supported functions. Smooth cutoffs and the weak product rule are available. (Compactly supported Sobolev functions extend by zero in every integer order, Compactly supported smooth functions are dense in W^{k,p}(R^n), Test function cutoffs and euclidean localization, Weak Leibniz rule with a smooth factor)

[F6]

A decreasing sequence of measurable subsets of a finite-measure set has measures tending to the measure of its intersection. (Continuity from above when one set has finite measure)

[F7]

The Sobolev class and its norm use the weak-derivative and $L^{p}$ conventions of Integer-order Sobolev spaces and their norms.

[F8]

Standard smooth mollifiers turn a Lipschitz cutoff into a smooth one; convolution with a unit-mass mollifier is smooth, and its Lipschitz bound does not exceed that of the original function. (The mollifier family generated by a unit-mass smooth bump, Convolution with a mollifier is smooth, and derivatives pass under the integral sign, Explicit compactly supported smooth cutoffs)

[F9]

The Sobolev and zero-boundary suppliers used above assume Countable Choice. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

Proof

1.1F1F2F7given

If $Ω = \emptyset$ , then $v = 0$ and the conclusion is immediate. Assume henceforth $Ω \neq \emptyset$ . Extend $v$ by zero to $\tilde{v}$ on $R^{d}$ and define $g_{i}$ to be its cellwise derivative on mesh cells and zero outside $\overline{Ω}$ . On each cell apply the simplex Green formula to $v φ e_{i}$ for an arbitrary $φ \in C_{c}^{\infty} (R^{d})$ . Fact [F1] gives the cell identity. Interior facet terms cancel by [F2]; boundary facet terms vanish by the zero-boundary clause of [F2]. Summing the finitely many cell identities gives $\int_{R^{d}} \tilde{v} \partial_{i} φ = - \int_{R^{d}} g_{i} φ .$ Both functions are in $L^{p}$ , so [F7] gives $\tilde{v} \in W^{1, p} (R^{d})$ with gradient $g$ .

1.2F5F8givenalgebra

Put $d_{Ω} (x) = dist (x, R^{d} ∖ Ω)$ , which is $1$ -Lipschitz. Choose a smooth scalar step $θ$ with $θ (s) = 0$ for $s \leq 1$ , $θ (s) = 1$ for $s \geq 2$ , and $0 \leq θ \leq 1$ . For $ϵ > 0$ set $ζ_{ϵ} (x) = θ (d_{Ω} (x) / ϵ)$ and convolve it with a nonnegative smooth mollifier of integral one supported in the ball of radius $ϵ / 4$ , obtaining $η_{ϵ}$ . Then $η_{ϵ}$ is smooth, supported in $Ω$ , equals zero where $d_{Ω} \leq 3 ϵ / 4$ , equals one where $d_{Ω} \geq 9 ϵ / 4$ , and $∥ \nabla η_{ϵ} ∥_{\infty} \leq ∥ θ^{'} ∥_{\infty} / ϵ$ . Since $Ω$ is bounded, its support is compact in $Ω$ .

2.1F2step 1.1given

Let $M$ be the maximum of the cellwise suprema of $∣ \nabla v ∣$ . Across every interior facet the polynomial traces agree; on every boundary facet they agree with zero. For a point $x$ in the mesh skeleton, its minimal mesh face is the unique mesh face whose relative interior contains $x$ ; the local star of that face means the union of the cells containing it. If the minimal face is a facet, there are exactly two incident cells at an interior facet and exactly one at a boundary facet. In the interior case their values at $x$ agree by the matching polynomial traces; in the boundary case the single cell value agrees with exterior zero by the zero boundary trace. For the remaining faces, let $c \geq 2$ be their codimension. The incident simplicial cones in their local star cut a small sphere in the normal space of the face, of dimension $c - 1$ , into a finite face-to-face triangulation of the local directions in $Ω$ . Within each connected component, top-dimensional pieces are connected by facet adjacency: perturb a path in that open component to avoid the finite union of codimension-two faces. At an interior point the direction set is the whole sphere, which is connected because $c \geq 2$ , so matching traces make all incident polynomial values agree. At a boundary point every component has a boundary direction; following facet adjacencies reaches a boundary facet, where the polynomial value is zero, so all cell limits in that component are zero. Thus in all cases the cellwise continuous representatives and the exterior zero fit continuously on the whole finite polyhedral partition. Restrict this continuous function to any line segment. The segment is divided by the finitely many cell facets into finitely many intervals; on each cell interval its derivative along the segment has magnitude at most $M$ , and on exterior intervals it is zero. Continuity at the interval endpoints then gives $∣ \tilde{v} (x) - \tilde{v} (y) ∣ \leq M ∣ x - y ∣$ for all $x, y$ . Thus $∣ v (x) ∣ \leq M dist (x, R^{d} ∖ Ω)$ .

3.1F5F6step 2.1step 1.2

The difference $v - η_{ϵ} v$ and its weak gradient are supported in $E_{ϵ} = {x \in Ω : d_{Ω} (x) < 9 ϵ / 4}$ . On this layer, step 2.1 gives $∣ v ∣ \leq (9 / 4) M ϵ$ , and step 1.2 gives $∣ v \nabla η_{ϵ} ∣ \leq (9 / 4) M ∥ θ^{'} ∥_{\infty}$ . Also $∣ \nabla v ∣ \leq M$ . For $n \in N_{0}$ , the sets $E_{1 / (n + 1)}$ decrease to the empty set and $∣ E_{1} ∣ \leq ∣ Ω ∣ < \infty$ , so [F6] gives $∣ E_{1 / (n + 1)} ∣ \to 0$ . These bounds show $∥ v - η_{1 / (n + 1)} v ∥_{W^{1, p} (Ω)} \to 0$ for finite $p$ .

4.1F3F5F9step 3.1

Each $η_{1 / (n + 1)} v$ has compact support in $Ω$ . By [F5] extend it by zero to $W^{1, p} (R^{d})$ and approximate there by $ϕ_{j} \in C_{c}^{\infty} (R^{d})$ . Choose a smooth cutoff $χ \in C_{c}^{\infty} (Ω)$ equal to one near the compact support of $η_{1 / (n + 1)} v$ . Then $χ ϕ_{j} \in C_{c}^{\infty} (Ω)$ and, by the weak product rule, $χ ϕ_{j} \to η_{1 / (n + 1)} v$ in $W^{1, p} (Ω)$ . A diagonal choice with step 3.1 proves $v \in W_{0}^{1, p} (Ω)$ by [F3].

5.1F4F9step 4.1∎

A bounded set lies in a slab of finite width $b - a$ . Apply [F4] to the membership from step 4.1, obtaining the displayed inequality with $C_{Ω} = C (p) (b - a)$ for the fixed exponent $p$ .

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Poincare–Wirtinger inequality on a reference simplex

Statement

Assume Countable Choice. For every $1 \leq p < \infty$ there is $C_{d, p} < \infty$ such that for all $u \in W^{1, p} ({\hat{S}}_{d}; R)$ , $∥ u - u_{{\hat{S}}_{d}} ∥_{L^{p} ({\hat{S}}_{d})} \leq C_{d, p} ∣ u ∣_{W^{1, p} ({\hat{S}}_{d})}, u_{{\hat{S}}_{d}} : = ∣ {\hat{S}}_{d} ∣^{- 1} \int_{{\hat{S}}_{d}} u d x .$

Facts & Assumptions

Given: Countable Choice, the bounded convex reference simplex and $1 \leq p < \infty$ .

[F1]

Smooth functions on a neighbourhood of the closed simplex are dense in $W^{1, p} ({\hat{S}}_{d})$ . (Smooth functions are dense in finite-order Sobolev spaces on a reference simplex)

[F2]

Along a segment, the fundamental theorem gives the integral of the directional derivative, bounded by segment length times the integral of $∣ \nabla u ∣$ . (If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz)

[F3]

Jensen bounds the $p$ th power of an average by the average of the $p$ th powers; Tonelli/Fubini permits the pair and parameter integrals; Holder controls integrals on the finite-measure simplex; affine change of variables has its determinant factor. (Jensen's integral inequality for a probability measure, Tonelli and Fubini for the completed product, with only almost-everywhere section measurability, Holder's inequality for integrals, including the endpoint cases, A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions)

[F4]

The smooth-density supplier [F1] assumes Countable Choice. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

Proof

1.1F2F3given

First take $u$ smooth. For $x, y \in {\hat{S}}_{d}$ , convexity keeps $z_{t} = t x + (1 - t) y$ in the simplex. The segment identity and $∣ x - y ∣ \leq h : = diam ({\hat{S}}_{d})$ give $∣ u (x) - u (y) ∣^{p} \leq h^{p} \int_{0}^{1} ∣ \nabla u (t x + (1 - t) y) ∣^{p} d t,$ by [F2] and Jensen on the unit interval. Jensen again gives $∣ u (x) - u_{{\hat{S}}_{d}} ∣^{p} \leq ∣ {\hat{S}}_{d} ∣^{- 1} \int_{{\hat{S}}_{d}} ∣ u (x) - u (y) ∣^{p} d y .$

2.1F3step 1.1algebra

Integrate the last bound over $x$ . For $t \geq 1 / 2$ , fix $y$ and use $x \mapsto t x + (1 - t) y$ ; its Jacobian is $t^{d}$ , and its image is contained in the simplex, so its integral is at most $2^{d} \int_{{\hat{S}}_{d}} ∣ \nabla u ∣^{p}$ . For $t \leq 1 / 2$ , fix $x$ and use $y \mapsto t x + (1 - t) y$ , whose inverse Jacobian is at most $2^{d}$ , giving the same bound. Tonelli [F3] therefore yields $∥ u - u_{{\hat{S}}_{d}} ∥_{p}^{p} \leq 2^{d} h^{p} ∥ \nabla u ∥_{p}^{p} .$ Taking $p$ th roots proves the estimate for smooth $u$ .

3.1F1F4step 2.1algebra∎

For arbitrary $u \in W^{1, p}$ choose smooth $u_{j} \to u$ in $W^{1, p}$ by [F1]. Holder on the finite-measure simplex gives $\int u_{j} \to \int u$ , so $(u_{j})_{{\hat{S}}_{d}} \to u_{{\hat{S}}_{d}}$ ; also $∥ u_{j} - u ∥_{p} \to 0$ and $∥ \nabla u_{j} ∥_{p} \to ∥ \nabla u ∥_{p}$ . Passing to the limit in the smooth inequality of step 2.1 proves the statement.

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Bramble–Hilbert approximation on a reference simplex

Statement

Assume Countable Choice. Let $1 \leq p < \infty$ , integer $r \geq 1$ , integer $s \geq 0$ , and real-valued $v \in W^{\max (r, s), p} ({\hat{S}}_{d})$ . There is a polynomial $q \in P_{r - 1} ({\hat{S}}_{d})$ , chosen by matching all means $\int_{{\hat{S}}_{d}} D^{α} q = \int_{{\hat{S}}_{d}} D^{α} v$ for $∣ α ∣ \leq r - 1$ , such that:

if $0 \leq s \leq r$ , then $∥ v - q ∥_{W^{s, p} ({\hat{S}}_{d})} \leq C ∣ v ∣_{W^{r, p} ({\hat{S}}_{d})}$ ;
if $s > r$ , then $∥ v - q ∥_{W^{s, p} ({\hat{S}}_{d})} \leq C \sum_{j = r}^{s} ∣ v ∣_{W^{j, p} ({\hat{S}}_{d})}$ .

The second clause assumes $v \in W^{s, p}$ ; constants depend only on $d, p, r, s$ .

Facts & Assumptions

Given: $1 \leq p < \infty$ , integers $r \geq 1$ and $s \geq 0$ , and $v \in W^{\max (r, s), p} ({\hat{S}}_{d})$ .

[F1]

The mean-zero Poincare–Wirtinger inequality holds on the reference simplex for every finite $p$ . (Poincare–Wirtinger inequality on a reference simplex)

[F2]

Sobolev classes provide all weak derivatives up to the specified integer order, and their norms are finite sums of derivative $L^{p}$ norms. (Integer-order Sobolev spaces and their norms)

[F3]

The mean-zero Poincare supplier [F1] assumes Countable Choice. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F4]

Holder's inequality on the finite-volume simplex sends $L^{p}$ derivatives into $L^{1}$ for finite $p$ . (Holder's inequality for integrals, including the endpoint cases)

Proof

1.1givenalgebra

The monomials $x^{α}$ with $∣ α ∣ \leq r - 1$ form a basis of $P_{r - 1}$ . Thus if $N$ is the number of these multi-indices, then $N = \dim P_{r - 1}$ . Define the linear moment map $M : P_{r - 1} \to R^{N}$ by $M (q)_{α} = ∣ {\hat{S}}_{d} ∣^{- 1} \int_{{\hat{S}}_{d}} D^{α} q$ . If every component is zero, take the largest degree $m$ of a nonzero homogeneous part of $q$ . For each $∣ α ∣ = m$ , $D^{α} q$ is the constant $α!$ times the coefficient of $x^{α}$ in that highest homogeneous part; its zero mean forces that coefficient to vanish. This contradicts the choice of $m$ . Repeating downward shows $q = 0$ , so $M$ is injective and, since its domain and codomain have the same finite dimension, bijective.

2.1F2F4step 1.1given

The vector of means of $D^{α} v$ , $∣ α ∣ \leq r - 1$ , is defined because each derivative is in $L^{p}$ on the finite-measure simplex and therefore in $L^{1}$ by [F4]. By bijectivity in step 1.1 there is a unique $q \in P_{r - 1}$ with those moments. For $w = v - q$ , each derivative $D^{α} w$ , $∣ α ∣ \leq r - 1$ , has zero mean; and every derivative of $q$ of order at least $r$ is zero.

3.1F1F2F3step 2.1

Apply [F1] to every mean-zero derivative of order $r - 1$ in step 2.1; its $L^{p}$ norm is bounded by the $L^{p}$ norms of derivatives of order $r$ of $v$ , since $D^{r} q = 0$ . Apply [F1] successively to derivatives of orders $r - 2, r - 3, \dots, 0$ , each of which also has zero mean; at each stage its norm is bounded by the next derivative order already controlled. Thus all derivatives of $w$ of order at most $r$ are bounded by $C ∣ v ∣_{W^{r, p}}$ . If $s \leq r$ , retaining only orders through $s$ proves the first clause.

4.1F2step 2.1step 3.1algebra∎

If $s > r$ , the same argument controls all derivative orders at most $r - 1$ by $C ∣ v ∣_{W^{r, p}}$ . For each $r \leq ∣ β ∣ \leq s$ , $D^{β} w = D^{β} v$ because $D^{β} q = 0$ . Adding the finitely many derivative bounds through order $s$ gives $∥ w ∥_{W^{s, p}} \leq C \sum_{j = r}^{s} ∣ v ∣_{W^{j, p}}$ , proving the second clause.

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Local interpolation estimates on shape-regular affine meshes

Statement

Assume Countable Choice. Let $k \geq 0$ , $1 \leq p < \infty$ , and let $({\hat{S}}_{d}, \hat{P}, \hat{Σ})$ be a reference finite element whose polynomial space contains $P_{k}$ . Assume its degrees of freedom are bounded on $W^{ℓ, p} ({\hat{S}}_{d})$ for an integer $ℓ \geq 1$ , let $r, m \in N_{0}$ , and let $\hat{I}$ be its interpolant. Thus $\hat{I} q = q$ for $q \in P_{k}$ . On a shape-regular affine mesh as in Matching affine simplicial meshes and shape regularity, for each cell $K$ with its orientation-preserving reference map $F_{K}$ define $I_{K} v = (\hat{I} (v \circ F_{K})) \circ F_{K}^{- 1} .$ For every such cell, if $ℓ \leq r \leq k + 1$ and $0 \leq m \leq r$ , then for $v \in W^{r, p} (K)$ , $∣ v - I_{K} v ∣_{W^{m, p} (K)} \leq C h_{K}^{r - m} ∣ v ∣_{W^{r, p} (K)} .$ If $ℓ > k + 1$ and $0 \leq m \leq k + 1$ , then for $v \in W^{ℓ, p} (K)$ , $∣ v - I_{K} v ∣_{W^{m, p} (K)} \leq C \sum_{n = k + 1}^{ℓ} h_{K}^{n - m} ∣ v ∣_{W^{n, p} (K)} .$ The constants are uniform over the mesh family and depend on the stated reference element, exponents and shape-regularity bound. For Lagrange elements in the existing mesh dimension range $d \geq 2$ , assume additionally the Axiom of Choice for the automatic nodal-boundedness application. Then Sobolev threshold for Lagrange point evaluations on a reference simplex supplies the boundedness hypothesis when $ℓ > d / p$ ; unisolvence of a particular node set remains a separate hypothesis.

Facts & Assumptions

Given: A shape-regular affine simplex family, finite $p$ , a unisolvent reference element, Countable Choice, and the stated degree-of-freedom boundedness on $W^{ℓ, p} ({\hat{S}}_{d})$ .

[F1]

Unisolvent degrees of freedom define a linear interpolation operator that reproduces every polynomial in its finite-element space. (Reference finite elements and unisolvent degrees of freedom, Unisolvent degrees of freedom define a unique reference interpolant)

[F2]

On the fixed reference polynomial space, every derivative norm is bounded by its coefficient or cell $L^{p}$ norm; hence bounded degrees of freedom give a bounded interpolant into every polynomial derivative norm. (Polynomial derivative and face norms are controlled on a reference simplex)

[F3]

For real-valued $v$ , Bramble–Hilbert supplies a polynomial $q$ of degree at most $r - 1$ with $∥ v - q ∥_{W^{m, p}} \leq C ∣ v ∣_{W^{r, p}}$ when $m \leq r$ , and for $q \in P_{k}$ with $v \in W^{ℓ, p}$ , $∥ v - q ∥_{W^{ℓ, p}} \leq C \sum_{n = k + 1}^{ℓ} ∣ v ∣_{W^{n, p}}$ when $ℓ > k + 1$ . (Bramble–Hilbert approximation on a reference simplex)

[F4]

Under affine pullback/pushforward, order- $s$ seminorms acquire $∥ A ∥^{s} ∣ \det A ∣^{- 1 / p}$ or $∥ A^{- 1} ∥^{s} ∣ \det A ∣^{1 / p}$ factors. (Affine pullback of weak derivatives and Sobolev seminorms)

[F5]

Shape regularity gives $∥ A_{K} ∥ ≍ h_{K}$ , $∥ A_{K}^{- 1} ∥ ≍ h_{K}^{- 1}$ and $\det A_{K} ≍ h_{K}^{d}$ . (Shape regularity controls affine simplex geometry)

[F6]

Under the additional Axiom of Choice and the existing dimension restriction $d \geq 2$ , Lagrange point evaluations are bounded when $ℓ > d / p$ . (Sobolev threshold for Lagrange point evaluations on a reference simplex) The generic estimate takes bounded degrees of freedom as a hypothesis.

[F7]

The Bramble–Hilbert supplier [F3] assumes Countable Choice through the reference Poincare–Wirtinger inequality. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F8]

For complex-valued Sobolev functions, real and imaginary parts and their weak derivatives are taken componentwise; the pointwise inequalities $∣ Re z ∣, ∣ Im z ∣ \leq ∣ z ∣ \leq ∣ Re z ∣ + ∣ Im z ∣$ combine the real-valued Sobolev estimates into the corresponding complex-valued estimates, with at most a fixed factor in the constant. (Integer-order Sobolev spaces and their norms)

Proof

1.1F1F2F6given

On the reference cell, write $\hat{I} w$ as the finite sum of its degree-of-freedom values times the dual shape functions. The assumed boundedness of each functional and [F2] bound $∣ \hat{I} w ∣_{W^{m, p}} \leq C ∥ w ∥_{W^{ℓ, p}}$ . For any reproduced $q$ , linearity gives $\hat{v} - \hat{I} \hat{v} = (\hat{v} - q) - \hat{I} (\hat{v} - q),$ so $∣ \hat{v} - \hat{I} \hat{v} ∣_{W^{m, p}} \leq ∣ \hat{v} - q ∣_{W^{m, p}} + C ∥ \hat{v} - q ∥_{W^{ℓ, p}} .$ For the automatic Lagrange application, under the additional Axiom of Choice, $d \geq 2$ , and $ℓ > d / p$ , [F6] verifies nodal-functional boundedness. For the generic estimate, boundedness is the explicit hypothesis, so this additional choice assumption is not used.

2.1F3F7F8step 1.1

If $ℓ \leq r \leq k + 1$ and $\hat{v}$ is real-valued, choose the Bramble–Hilbert polynomial $q \in P_{r - 1} \subseteq P_{k}$ from [F3]. If $\hat{v}$ is complex-valued, apply the first clause of [F3] to $Re \hat{v}$ and $Im \hat{v}$ separately, obtaining $q_{R}, q_{I} \in P_{r - 1}$ , and set $q = q_{R} + i q_{I}$ . In either case $q$ is reproduced by $\hat{I}$ . Since $m \leq r$ and $ℓ \leq r$ , [F3] bounds both terms in step 1.1 for the real case; for the complex case, [F8] combines the two real estimates and changes only the constant. Thus $∣ \hat{v} - \hat{I} \hat{v} ∣_{W^{m, p}} \leq C ∣ \hat{v} ∣_{W^{r, p}}$ .

2.2F3F7F8step 1.1

If $ℓ > k + 1$ and $\hat{v}$ is real-valued, choose the Bramble–Hilbert polynomial $q \in P_{k}$ from [F3]. If $\hat{v}$ is complex-valued, apply the second clause of [F3] with $r = k + 1$ , $s = ℓ$ to its real and imaginary parts, obtaining $q_{R}, q_{I} \in P_{k}$ , and set $q = q_{R} + i q_{I}$ . In either case $q$ is reproduced by $\hat{I}$ . Then [F3] bounds the $W^{ℓ, p}$ norm of $\hat{v} - q$ by $C \sum_{n = k + 1}^{ℓ} ∣ \hat{v} ∣_{W^{n, p}}$ ; because $m \leq k + 1 < ℓ$ , the same controls its $W^{m, p}$ norm. For complex $\hat{v}$ , [F8] combines the component estimates with at most a fixed change in $C$ . Step 1.1 gives $∣ \hat{v} - \hat{I} \hat{v} ∣_{W^{m, p}} \leq C \sum_{n = k + 1}^{ℓ} ∣ \hat{v} ∣_{W^{n, p}} .$

3.1F4F5F7step 2.1algebra

For $\hat{v} = v \circ F_{K}$ , use [F4] to transform the physical error to the reference cell and the reference derivatives of $v$ back to the physical cell. The product of the two determinant factors cancels; the remaining matrix factors are bounded by $C ∥ A_{K}^{- 1} ∥^{m} ∥ A_{K} ∥^{r} \leq C h_{K}^{r - m}$ by [F5]. This gives the first physical estimate from step 2.1.

4.1F2F4F5F7step 2.2algebra∎

In the second case, apply the same transformation separately to each order $n$ in step 2.2. Its matrix factor is bounded by $C ∥ A_{K}^{- 1} ∥^{m} ∥ A_{K} ∥^{n} \leq C h_{K}^{n - m}$ by [F5]; summing over $k + 1 \leq n \leq ℓ$ gives the second physical estimate. The constants in [F2]–[F7] are uniform over the fixed reference element and the shape-regular family, proving uniformity.

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Polynomial inverse and discrete trace inequalities

Statement

Assume Countable Choice. Fix polynomial degree $k \geq 0$ , $1 \leq p, q \leq \infty$ , and integers $0 \leq m \leq ℓ \leq k$ . Let the shape-regular affine simplex family be as in Matching affine simplicial meshes and shape regularity. For every cell $K$ in that family, there is a constant independent of cell size such that, for every $v \in P_{k} (K)$ and every facet $F \subset \partial K$ , $∣ v ∣_{W^{ℓ, p} (K)} \leq C h_{K}^{m - ℓ + d (1 / p - 1 / q)} ∣ v ∣_{W^{m, q} (K)},$ $∥ v ∥_{L^{p} (F)} \leq C h_{K}^{(d - 1) / p - d / q} ∥ v ∥_{L^{q} (K)} .$ Use $1 / \infty = 0$ . Constants may depend on $d, k, p, q$ and the family shape regularity bound, but not on $K$ or $h_{K}$ .

Facts & Assumptions

Given: Countable Choice, the fixed reference simplex, a shape-regular affine mesh family, polynomial degree $k$ , the displayed exponents, and a cell polynomial $v$ .

[F1]

On the fixed finite-dimensional polynomial space, $∣ v ∣_{W^{ℓ, p}} \leq C ∣ v ∣_{W^{m, q}}$ when $m \leq ℓ$ , and every facet $L^{p}$ norm is bounded by the cell $L^{q}$ norm, for all $p, q \in [1, \infty]$ . (Polynomial derivative and face norms are controlled on a reference simplex)

[F2]

Under an affine map, order- $r$ Sobolev seminorms acquire factors $∥ A ∥^{r} ∣ \det A ∣^{- 1 / q}$ on pullback and $∥ A^{- 1} ∥^{r} ∣ \det A ∣^{1 / p}$ on pushforward. Its oriented face area vector is $\det (A) A^{- T} \hat{n} d \hat{S}$ . (Affine pullback of weak derivatives and Sobolev seminorms, Gauss–Green formula on an affine simplex)

[F3]

Shape regularity gives $∥ A_{K} ∥ ≍ h_{K}$ , $∥ A_{K}^{- 1} ∥ ≍ h_{K}^{- 1}$ , $\det A_{K} ≍ h_{K}^{d}$ , and each facet Jacobian is comparable to $h_{K}^{d - 1}$ . (Shape regularity controls affine simplex geometry)

[F4]

The reference polynomial norm result [F1] uses Sobolev norms whose definition assumes Countable Choice. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

Proof

1.1F1given

Pull $v$ back to $\hat{v} = v \circ F_{K} \in P_{k} ({\hat{S}}_{d})$ . By [F1], $∣ \hat{v} ∣_{W^{ℓ, p}} \leq C ∣ \hat{v} ∣_{W^{m, q}}$ and $∥ \hat{v} ∥_{L^{p} (\hat{F})} \leq C ∥ \hat{v} ∥_{L^{q}}$ for each reference facet; finitely many facets allow one constant.

2.1F2F3F4step 1.1algebra∎

Apply the affine derivative estimates [F2] in both directions and then [F3]. For the cell inequality the total factor is bounded by $∥ A_{K}^{- 1} ∥^{ℓ} ∥ A_{K} ∥^{m} ∣ \det A_{K} ∣^{1 / p - 1 / q} \leq C h_{K}^{m - ℓ + d (1 / p - 1 / q)} .$ For the face inequality it is bounded by the physical face Jacobian to power $1 / p$ times the inverse cell Jacobian to power $1 / q$ , namely $C h_{K}^{(d - 1) / p - d / q}$ . These are the two claimed estimates.

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Covariant and contravariant Piola maps and H(div)/H(curl) spaces

Definition

Assume Countable Choice. Let $d \geq 2$ , take $\hat{K} = {\hat{S}}_{d}$ as in Matching affine simplicial meshes and shape regularity, and let $K = F (\hat{K})$ under an affine cell map $F (\hat{x}) = A \hat{x} + b$ with $\det A > 0$ . Weak derivatives use the signed test identity of Weak derivative of a locally integrable function. For $d \geq 2$ define $H (div, K) : = {v \in L^{2} (K; R^{d}) : div v \in L^{2} (K)},$ where $div v = \sum_{i} D_{i} v_{i}$ weakly, meaning $\int_{K} v \cdot \nabla ϕ = - \int_{K} (div v) ϕ$ for every $ϕ \in C_{c}^{\infty} (K)$ . For $d = 2$ , define the weak scalar curl by $\int_{K} (curl v) ϕ = \int_{K} (v_{1} \partial_{2} ϕ - v_{2} \partial_{1} ϕ)$ for every $ϕ \in C_{c}^{\infty} (K)$ . For $d = 3$ , define the weak vector curl by $\int_{K} (curl v) \cdot ψ = \int_{K} v \cdot curl ψ$ for every $ψ \in C_{c}^{\infty} (K; R^{3})$ . In dimensions $d = 2, 3$ set $H (curl, K) : = {v \in L^{2} (K; R^{d}) : curl v \in L^{2} (K; R^{d_{*}})},$ with $d_{*} = 1$ for $d = 2$ and $d_{*} = 3$ for $d = 3$ .

The covariant Piola pushforward of a reference vector field is $(P_{F}^{c u r l} \hat{v}) (F (\hat{x})) : = A^{- T} \hat{v} (\hat{x}),$ and the contravariant Piola pushforward is $(P_{F}^{d i v} \hat{v}) (F (\hat{x})) : = \frac{1}{\det A} A \hat{v} (\hat{x}) .$ The scalar pullback is $u \mapsto u \circ F$ . The exact mapping properties, weak differential identities, and oriented trace-moment transformations are established in the corresponding affine Piola theorem below.

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Affine Piola maps commute with differential operators and preserve moments

Statement

Assume Countable Choice. Let $d = 2$ or $3$ and let $F (\hat{x}) = A \hat{x} + b$ map a reference simplex to an affine cell with $\det A > 0$ . The contravariant Piola map $P_{F}^{d i v} \hat{v} = (\det A)^{- 1} A \hat{v} \circ F^{- 1}$ maps $H (div, {\hat{S}}_{d})$ into $H (div, K)$ and $div (P_{F}^{d i v} \hat{v}) = (\det A)^{- 1} (\overset{}{div} \hat{v}) \circ F^{- 1} .$ The covariant Piola map $P_{F}^{c u r l} \hat{v} = A^{- T} \hat{v} \circ F^{- 1}$ maps $H (curl, {\hat{S}}_{d})$ into $H (curl, K)$ ; in $d = 2$ its scalar curl is $(\det A)^{- 1} (\overset{}{curl} \hat{v}) \circ F^{- 1}$ , and in $d = 3$ its vector curl is $(\det A)^{- 1} A (\overset{}{curl} \hat{v}) \circ F^{- 1}$ . For every $\hat{u} \in W^{1, 2} ({\hat{S}}_{d}; R)$ , the scalar pullback $u = \hat{u} \circ F^{- 1}$ belongs to $W^{1, 2} (K; R)$ and its gradient satisfies $\nabla u = P_{F}^{c u r l} (\hat{\nabla} \hat{u}) .$ For smooth fields, contravariant normal fluxes satisfy $v \cdot n d S = \hat{v} \cdot \hat{n} d \hat{S}$ ; covariant tangential pullbacks satisfy $v (F (\hat{x})) \cdot A \hat{τ} = \hat{v} (\hat{x}) \cdot \hat{τ}$ for every reference tangent $\hat{τ}$ . Thus edge line integrals and normal-flux moments are preserved, and face tangential moments transform with this pullback and the induced facet surface Jacobian. These formulas record the chosen orientation; reversing a facet orientation reverses its signed moment.

Facts & Assumptions

Given: Countable Choice, the affine map, $\det A > 0$ , $d \in {2, 3}$ , and the weak graph spaces and Piola maps of Covariant and contravariant Piola maps and H(div)/H(curl) spaces.

[F1]

Weak derivatives pull back through an affine map by the constant matrix chain rule, and affine change of variables multiplies volume by $\det A$ . (Affine pullback of weak derivatives and Sobolev seminorms)

[F2]

The cofactor area-vector identity is $n d S = \det (A) A^{- T} \hat{n} d \hat{S}$ , with outward orientation preserved when $\det A > 0$ . (Gauss–Green formula on an affine simplex)

[F3]

Lebesgue change of variables holds for nonnegative measurable integrands; apply it to positive and negative parts for integrable signed pairings. (A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions)

[F4]

The affine pullback and change-of-variables suppliers assume Countable Choice. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F5]

Applying the affine weak-derivative transformation to $F^{- 1}$ gives $D_{x_{i}} (\hat{u} \circ F^{- 1}) = \sum_{j} (A^{- 1})_{j i} (D_{{\hat{x}}_{j}} \hat{u}) \circ F^{- 1}$ for $\hat{u} \in W^{1, 2} ({\hat{S}}_{d}; R)$ . (Affine pullback of weak derivatives and Sobolev seminorms)

Proof

1.1F1F3F4given

For $\hat{v} \in H (div, {\hat{S}}_{d})$ and $ϕ \in C_{c}^{\infty} (K)$ , put $\hat{ϕ} = ϕ \circ F$ . Then [F1] and the determinant substitution give $\int_{K} (P_{F}^{d i v} \hat{v}) \cdot \nabla ϕ d x = \int_{{\hat{S}}_{d}} \hat{v} \cdot \nabla \hat{ϕ} d \hat{x} = - \int_{{\hat{S}}_{d}} (\overset{}{div} \hat{v}) \hat{ϕ} d \hat{x} .$ The final expression equals $- \int_{K} [(\det A)^{- 1} \overset{}{div} \hat{v} \circ F^{- 1}] ϕ d x$ . This is the distributional divergence identity.

1.2F1algebra

For smooth $\hat{v}$ , differentiate $v = A^{- T} \hat{v} \circ F^{- 1}$ componentwise. In three dimensions, write $C = A^{- 1}$ . For any $z \in R^{3}$ and indices $m, ℓ$ , the scalar triple product gives $z \cdot ((C^{T} e_{m}) \times (C^{T} e_{ℓ})) = \det [z, C^{T} e_{m}, C^{T} e_{ℓ}] = \det (C) \det [C^{- T} z, e_{m}, e_{ℓ}] = \det (C) z \cdot (C^{- 1} (e_{m} \times e_{ℓ})) .$ Since this holds for every $z$ , comparing components yields $ϵ_{i j k} C_{m j} C_{ℓ k} = (\det A)^{- 1} A_{i r} ϵ_{r m ℓ}$ . The chain rule then gives ${curl}_{x} v = (\det A)^{- 1} A ({curl}_{\hat{x}} \hat{v}) \circ F^{- 1}$ . In two dimensions write $A = (\begin{array}{cc} a & b \\ c & d \end{array})$ and $Δ = a d - b c > 0$ . Then $v_{1} = (d {\hat{v}}_{1} - c {\hat{v}}_{2}) / Δ$ , $v_{2} = (- b {\hat{v}}_{1} + a {\hat{v}}_{2}) / Δ$ , $\partial_{x_{1}} = (d \partial_{{\hat{x}}_{1}} - c \partial_{{\hat{x}}_{2}}) / Δ$ , and $\partial_{x_{2}} = (- b \partial_{{\hat{x}}_{1}} + a \partial_{{\hat{x}}_{2}}) / Δ$ . Substituting these four expressions and cancelling the two mixed terms gives $\partial_{x_{1}} v_{2} - \partial_{x_{2}} v_{1} = (\det A)^{- 1} (\partial_{{\hat{x}}_{1}} {\hat{v}}_{2} - \partial_{{\hat{x}}_{2}} {\hat{v}}_{1}) \circ F^{- 1}$ .

2.1F1F3F4step 1.1algebra

The physical field and candidate divergence in step 1.1 lie in $L^{2} (K)$ : affine change of variables makes their squared norms finite constant multiples of the reference squared norms, and multiplication by the fixed matrix $A / (\det A)$ preserves finiteness. The test identity in step 1.1 therefore gives membership in $H (div, K)$ and the asserted divergence formula.

2.2F1F3F4step 1.2givenalgebra

For $d = 3$ , take $ψ \in C_{c}^{\infty} (K; R^{3})$ and set $\hat{ψ} = A^{T} (ψ \circ F)$ . The component calculation in step 1.2 gives ${curl}_{\hat{x}} \hat{ψ} = \det (A) A^{- 1} ({curl}_{x} ψ) \circ F$ . Change variables and use the weak curl identity on the reference cell: $\int_{K} v \cdot curl ψ d x = \int_{{\hat{S}}_{d}} \hat{v} \cdot curl \hat{ψ} d \hat{x} = \int_{{\hat{S}}_{d}} (\overset{}{curl} \hat{v}) \cdot \hat{ψ} d \hat{x} = \int_{K} [(\det A)^{- 1} A (\overset{}{curl} \hat{v}) \circ F^{- 1}] \cdot ψ d x .$ For $d = 2$ , take $ϕ \in C_{c}^{\infty} (K)$ and set $\hat{ϕ} = ϕ \circ F$ . The determinant identity $\det (A) \det (A^{- T} \hat{v}, A^{- T} \nabla \hat{ϕ}) = \det (\hat{v}, \nabla \hat{ϕ})$ transforms the weak scalar-curl pairing into $\int_{{\hat{S}}_{d}} ({\hat{v}}_{1} \partial_{2} \hat{ϕ} - {\hat{v}}_{2} \partial_{1} \hat{ϕ})$ , which by the weak reference curl identity equals $\int_{{\hat{S}}_{d}} (\overset{}{curl} \hat{v}) \hat{ϕ}$ and hence the pairing with the asserted physical scalar curl. In both dimensions the mapped field and candidate curl lie in $L^{2}$ by the fixed-matrix and determinant bounds. Thus the weak definition gives the claimed $H (curl)$ mapping.

2.3F2F4step 1.1step 1.2algebra

The normal flux identity follows by dotting $v = (\det A)^{- 1} A \hat{v}$ with the area vector [F2]: $v \cdot n d S = (\det A)^{- 1} A \hat{v} \cdot \det (A) A^{- T} \hat{n} d \hat{S} = \hat{v} \cdot \hat{n} d \hat{S} .$ For a reference tangent $\hat{τ}$ , $v (F) \cdot A \hat{τ} = (A^{- T} \hat{v}) \cdot A \hat{τ} = \hat{v} \cdot \hat{τ}$ . Along an edge this equality is the line-form identity $v \cdot d x = \hat{v} \cdot d \hat{x}$ . On a facet, integration against a pulled-back tangent vector and scalar weight gives $\int_{F} v \cdot (A \hat{τ}) (\hat{q} \circ F^{- 1}) d S = \int_{\hat{F}} (\hat{v} \cdot \hat{τ}) \hat{q} J_{F} d \hat{S},$ where $J_{F}$ is the affine facet surface Jacobian. The normal-flux moment with scalar weight $\hat{q}$ equals $\int_{\hat{F}} (\hat{v} \cdot \hat{n}) \hat{q} d \hat{S}$ . The signs are those of the selected outward facet orientation; reversing it reverses the signed normal moment.

3.1F1F5algebra∎

For $\hat{u} \in W^{1, 2} ({\hat{S}}_{d}; R)$ , apply [F5] to $F^{- 1}$ . For each $i$ , $D_{x_{i}} (\hat{u} \circ F^{- 1}) = \sum_{j} (A^{- 1})_{j i} (D_{{\hat{x}}_{j}} \hat{u}) \circ F^{- 1} .$ In vector notation this is $\nabla_{x} (\hat{u} \circ F^{- 1}) = A^{- T} (\hat{\nabla} \hat{u}) \circ F^{- 1} = P_{F}^{c u r l} (\hat{\nabla} \hat{u}),$ which proves the scalar-gradient identity.

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Affine Piola transport preserves commuting interpolation diagrams

Statement

Let ${\hat{V}}_{j}$ and $V_{j}$ be reference and physical spaces related by invertible affine Piola maps $P_{j} : {\hat{V}}_{j} \to V_{j}$ . Let ${\hat{D}}_{j}$ and $D_{j}$ be the corresponding differential operators, and assume the proved affine identities $D_{j} P_{j} = P_{j + 1} {\hat{D}}_{j}$ from Affine Piola maps commute with differential operators and preserve moments. Suppose reference interpolation operators ${\hat{I}}_{j}$ are defined on spaces containing the relevant inputs and satisfy ${\hat{D}}_{j} {\hat{I}}_{j} = {\hat{I}}_{j + 1} {\hat{D}}_{j}$ there. Define the physical interpolants by Piola conjugation $I_{j} = P_{j} {\hat{I}}_{j} P_{j}^{- 1}$ . Then $D_{j} I_{j} = I_{j + 1} D_{j}$ on every input for which these compositions are defined. This is conditional on the reference commuting identity; it does not assert that arbitrary unisolvent degrees of freedom commute.

Facts & Assumptions

Given: Unisolvent reference interpolants as in Reference finite elements and unisolvent degrees of freedom and Unisolvent degrees of freedom define a unique reference interpolant, invertible affine Piola maps, and the reference commuting identities in the statement.

[F1]

The physical and reference differential operators satisfy $D_{j} P_{j} = P_{j + 1} {\hat{D}}_{j}$ on their domains. (Affine Piola maps commute with differential operators and preserve moments)

[F2]

The interpolation operator defined by the dual shape functions is a linear map on its declared domain. (Unisolvent degrees of freedom define a unique reference interpolant)

Proof

1.1F1givenalgebra

By definition $I_{j} = P_{j} {\hat{I}}_{j} P_{j}^{- 1}$ . Using [F1] and the assumed reference commutation, $D_{j} I_{j} = D_{j} P_{j} {\hat{I}}_{j} P_{j}^{- 1} = P_{j + 1} {\hat{D}}_{j} {\hat{I}}_{j} P_{j}^{- 1} = P_{j + 1} {\hat{I}}_{j + 1} {\hat{D}}_{j} P_{j}^{- 1} .$

2.1F1F2step 1.1algebra∎

Apply [F1] to $P_{j}^{- 1}$ , equivalently ${\hat{D}}_{j} P_{j}^{- 1} = P_{j + 1}^{- 1} D_{j}$ . Substitution into step 1.1 gives $D_{j} I_{j} = P_{j + 1} {\hat{I}}_{j + 1} P_{j + 1}^{- 1} D_{j} = I_{j + 1} D_{j},$ which is the claimed physical commuting identity. The only interpolation property used was the explicitly assumed reference identity, proving the conditional scope.

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Sobolev threshold for Lagrange point evaluations on a reference simplex

Statement

Assume the Axiom of Choice. Let $d \geq 2$ , let $N \in N_{0}$ , choose $x_{1}, \dots, x_{N} \in \overline{{\hat{S}}_{d}}$ , and let $ℓ \geq 1$ be an integer. If $1 \leq p < \infty$ and $ℓ > d / p$ , then each point evaluation has a unique continuous-representative value and $∣ u (x_{j}) ∣ \leq C ∥ u ∥_{W^{ℓ, p} ({\hat{S}}_{d})} .$ If $p = \infty$ , the same conclusion holds for every $ℓ \geq 1$ . Consequently any finite set of Lagrange nodal evaluations is bounded on the stated Sobolev space and defines a well-defined degree-of-freedom functional there. No boundedness at $ℓ = d / p$ is asserted.

Facts & Assumptions

Given: The Axiom of Choice, $d \geq 2$ , integer $ℓ \geq 1$ , the reference simplex, and the exponent in the appropriate branch.

[A1]

The Axiom of Choice states that every family of nonempty sets has a choice function. (The Axiom of Choice)

[F1]

The reference simplex is a bounded extension domain for every integer Sobolev order and exponent. (A reference simplex is an integer-order Sobolev extension domain)

[F2]

On a bounded extension domain, if $k p > d$ , higher-order Sobolev embedding gives a continuous representative and a norm bound; its proof assumes the Axiom of Choice. (Higher-order Sobolev embedding)

[F3]

For finite $p$ , the Sobolev norm is the $ℓ^{p}$ norm of the finite vector of derivative $L^{p}$ norms; for $p = \infty$ it is their maximum. (Integer-order Sobolev spaces and their norms)

[F4]

Holder's inequality on the finite-measure simplex sends each bounded $L^{\infty}$ derivative into every finite $L^{q}$ space with the expected measure factor. (Holder's inequality for integrals, including the endpoint cases)

[F5]

The reference finite-element convention requires a unisolvent list of linear functionals on its finite-dimensional space and chosen bounded extensions to the interpolation domain. Thus, for nodal degrees of freedom of such an element, the Sobolev evaluation maps must extend the polynomial evaluations; unisolvence is a separate hypothesis. (Reference finite elements and unisolvent degrees of freedom)

Proof

1.1A1F1F2given

Suppose $p < \infty$ and $ℓ > d / p$ . Then $ℓ p > d$ , so apply [F2] with $k = ℓ$ , $n = d$ , and extension domain [F1]. It gives a continuous representative on the closed simplex and a bound for its supremum by $C ∥ u ∥_{W^{ℓ, p}}$ . Two continuous representatives of the same almost-everywhere class agree everywhere: if their difference were nonzero at a point of the closed simplex, continuity on the closure and the fact that the simplex is the closure of its interior would give a nonempty interior ball on which the difference stays nonzero, contradicting almost-everywhere equality. Thus every $u (x_{j})$ is uniquely defined and obeys the displayed bound.

2.1A1F1F2F3F4F5algebra∎

If $p = \infty$ , choose any finite $q > d$ . Since the simplex has finite volume, [F4] gives $∥ D^{α} u ∥_{L^{q}} \leq ∣ {\hat{S}}_{d} ∣^{1 / q} ∥ D^{α} u ∥_{L^{\infty}}$ for $∣ α ∣ \leq 1$ , so $W^{ℓ, \infty} \subset W^{1, q}$ continuously. The same simplex extension result [F1] and embedding [F2], now with $k = 1$ and $q > d$ , give a unique continuous representative and the required bound by $∥ u ∥_{W^{ℓ, \infty}}$ . In either branch, evaluation is linear because a linear combination of continuous representatives is the unique continuous representative of the corresponding Sobolev class. On a polynomial it is the ordinary nodal value, so these bounded maps extend any prescribed nodal degrees of freedom as required by [F5], without asserting unisolvence of an arbitrary node list; for $N \geq 1$ , the nodes have a common bound by taking the maximum of their constants, while for $N = 0$ the assertion is vacuous and any finite constant, such as $C = 1$ , suffices.

5 · Examples, counterexamples and false statements

None yet.