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Well-posedness and energy stability of the semidiscrete FEM system

Statement

Assume Countable Choice. Let $d \geq 2$ and let $Ω \subset R^{d}$ be a bounded polyhedral open set, let $T_{h}$ be a finite matching affine simplicial mesh of $Ω$ , and choose $k \in N_{0}$ . Let $V = W_{0}^{1, 2} (Ω; R)$ with its $W^{1, 2}$ norm, $H = L^{2} (Ω; R)$ , and let $V^{*}$ be the continuous dual of $V$ ; these spaces form the separable real evolution triple proved in The zero-boundary Sobolev energy space is a separable Hilbert space. Let $V_{h}$ be the scalar degree-at-most- $k$ conforming polynomial subspace on $T_{h}$ whose polynomial traces vanish on every boundary facet, as defined by Broken and conforming polynomial spaces on a matching mesh. Suppose the real bilinear form $a (t; \cdot, \cdot)$ on $V$ is scalar measurable and there is one null set outside which it is bounded by $M$ and coercive by $α > 0$ on $V$ . Let $f \in L^{2} (0, T; V^{*})$ and $u_{h}^{0} \in V_{h}$ . Then there is a unique $u_{h} \in H^{1} (0, T; V_{h})$ with $u_{h} (0) = u_{h}^{0}$ satisfying $((u_{h})^{'} (t), v_{h})_{H} + a (t; u_{h} (t), v_{h}) = ⟨ f (t), v_{h} ⟩_{V^{*}, V} (v_{h} \in V_{h})$ for almost every $t$ . For every $t \in [0, T]$ , $∥ u_{h} (t) ∥_{H}^{2} + α \int_{0}^{t} ∥ u_{h} (s) ∥_{V}^{2} d s \leq ∥ u_{h}^{0} ∥_{H}^{2} + α^{- 1} \int_{0}^{t} ∥ f (s) ∥_{V^{*}}^{2} d s .$

Facts & Assumptions

Given: Countable Choice, a bounded polyhedral open set, a finite matching affine mesh and its scalar zero-boundary conforming polynomial space, the $W^{1, 2}$ -norm evolution triple, the uniformly bounded/coercive measurable form, $f \in L^{2} (0, T; V^{*})$ , and $u_{h}^{0} \in V_{h}$ .

[F1]

On a finite matching affine mesh, the zero-boundary conforming polynomial space from Matching affine simplicial meshes and shape regularity is a finite-dimensional algebraic subspace of the broken polynomial product (Broken and conforming polynomial spaces on a matching mesh).

[F2]

If its polynomial traces vanish on boundary facets and match on interior facets, that subspace lies in $W_{0}^{1, 2} (Ω)$ ; this is a closed subspace of the ambient evolution space under the present identification (Zero-boundary conforming polynomials lie in W0 1p and satisfy Poincare).

[F3]

On any finite-dimensional $V_{h} \subset V$ , the measurable-coefficient Galerkin ODE has a unique absolutely continuous solution and the displayed energy estimate (Finite-dimensional parabolic Galerkin ODEs with measurable coefficients).

[F4]

For bounded polyhedral $Ω$ , $V = W_{0}^{1, 2} (Ω; R)$ with the $W^{1, 2}$ norm and $H = L^{2} (Ω; R)$ form a separable real evolution triple (The zero-boundary Sobolev energy space is a separable Hilbert space).

[A1]

Countable Choice is assumed by the zero-boundary Sobolev supplier (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

1.1F1F2F4A1given

By [F1], $V_{h}$ is finite-dimensional. Its zero-boundary trace condition and [F2] give $V_{h} \subset V$ , while [F4] verifies that the stated $V, H, V^{*}$ are a separable evolution triple. Countable Choice is retained exactly as in [A1].

2.1F3step 1.1given∎

Restriction of each $F (t) \in V^{*}$ to $V_{h}$ is a bounded functional, and the restriction map $V^{*} \to V_{h}^{*}$ is continuous. Thus the restricted load is strongly measurable and square-integrable. Apply [F3] to this $V_{h}$ and the restricted form/load. It gives a unique $u_{h}$ satisfying the stated semidiscrete equation and energy estimate.

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Sources

Christian Clason, Introduction to Finite Element Methods, lecture notes (standard reference, not scraped)
Long Chen, Finite Element Methods for Parabolic Equations (standard reference, not scraped)