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

Statement

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

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

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

Facts & Assumptions

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

[F1]

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

[F2]

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

[F3]

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

[A1]

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

Proof

1.1A1F1F3given

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

2.1F2step 1.1algebra∎

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

Depends on

Used by

An analytic box solution has exponential Galerkin p convergence Example

Dependency tree · two levels

26 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jean-Luc Guermond, MATH 661 Fall 2025, Part VI, Basic error analysis (standard reference, not scraped)
Martin Vohralík, Finite Element Methods (standard reference, not scraped)