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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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Sources

Jean-Luc Guermond, MATH 661 Fall 2025, Chapter 26, Basic error analysis (standard reference, not scraped)
Martin Vohralík, Finite Element Methods, Chapter 6, hp finite elements (standard reference, not scraped)