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Semidiscrete parabolic energy and L2 rates from elliptic approximation

Statement

Assume Countable Choice and the real scalar finite-element setting of Primal H¹ approximation on shape-regular affine finite-element meshes, on a shape-regular affine mesh family with $h \leq 1$ . Let $V = H_{0}^{1} (Ω)$ with its $H^{1}$ norm, $H = L^{2} (Ω)$ , and let $a$ be a time-independent bounded coercive form with constants $M, α$ . Let $R_{h}$ be its Ritz projection and let $u_{h}$ be the semidiscrete solution initialized by $u_{h} (0) = R_{h} u (0)$ . Assume the continuous solution obeys $u \in H^{1} (0, T; V)$ , $u \in L^{\infty} (0, T; W^{r + 1, 2} (Ω) \cap V), u^{'} \in L^{2} (0, T; W^{r + 1, 2} (Ω) \cap V),$ for an integer $r \geq 1$ . For almost every $t$ , assume the adjoint solution and approximation inequalities in The h^(r+s) L² rate under primal and dual approximation hold for each target $w = u (t)$ and $w = u^{'} (t)$ with common $s \in (0, 1]$ , $C_{D}$ , and $C_{R}$ , uniformly in $h, t$ . Let $C_{F E}$ be the primal interpolation constant in that corollary and put $C_{A} = \frac{M^{2}}{α} C_{F E} C_{D} C_{R}, c_{i} = ∥ V ↪ H ∥ .$ Then $ρ = R_{h} u - u$ and $θ = u_{h} - R_{h} u$ satisfy $∥ ρ ∥_{L^{\infty} (0, T; V)} \leq \frac{M}{α} C_{F E} h^{r} ∥ u ∥_{L^{\infty} (0, T; W^{r + 1, 2})},$ $∥ ρ ∥_{L^{\infty} (0, T; H)} \leq C_{A} h^{r + s} ∥ u ∥_{L^{\infty} (0, T; W^{r + 1, 2})}, ∥ ρ^{'} ∥_{L^{2} (0, T; H)} \leq C_{A} h^{r + s} ∥ u^{'} ∥_{L^{2} (0, T; W^{r + 1, 2})},$ and $∥ θ ∥_{L^{\infty} (0, T; H)} \leq \frac{c_{i} C_{A}}{\sqrt{α}} h^{r + s} ∥ u^{'} ∥_{L^{2} (0, T; W^{r + 1, 2})}, ∥ θ ∥_{L^{2} (0, T; V)} \leq \frac{c_{i} C_{A}}{α} h^{r + s} ∥ u^{'} ∥_{L^{2} (0, T; W^{r + 1, 2})} .$ Consequently, $∥ u_{h} - u ∥_{L^{\infty} (0, T; H)}$ is bounded by the sum of the two $L^{\infty} (0, T; H)$ bounds for $ρ, θ$ , and $∥ u_{h} - u ∥_{L^{2} (0, T; V)} \leq \sqrt{T} \frac{M}{α} C_{F E} h^{r} ∥ u ∥_{L^{\infty} (0, T; W^{r + 1, 2})} + \frac{c_{i} C_{A}}{α} h^{r + s} ∥ u^{'} ∥_{L^{2} (0, T; W^{r + 1, 2})} .$ In particular, the full-adjoint branch $s = 1$ gives the conditional semidiscrete max-in-time rate $h^{r + 1}$ in $L^{\infty} (H)$ ; it requires both stated dual hypotheses and does not assert full adjoint regularity on an arbitrary polyhedral domain.

Facts & Assumptions

Given: Countable Choice; the exact shape-regular affine finite-element interpolation hypotheses; a real time-independent bounded/coercive form; the continuous and semidiscrete solutions with exact initial Ritz projection; the spatial regularity of $u, u^{'}$ ; and the two uniform adjoint conditions for each of those targets.

[A1]

Countable Choice is assumed in the Cea and adjoint Galerkin suppliers (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[A2]

The two adjoint regularity and approximation inequalities hold uniformly for the elliptic Ritz errors with targets $u (t)$ and $u^{'} (t)$ , with the common constants $C_{D}, C_{R}$ in Statement (The h^(r+s) L² rate under primal and dual approximation).

[A3]

The displayed time-dependent spatial Sobolev regularity assumptions on $u$ and $u^{'}$ hold.

[F6]

$W_{0}^{1, 2} (Ω)$ with its usual $H^{1}$ norm, $L^{2} (Ω)$ , and the dual form a separable real evolution triple (The zero-boundary Sobolev energy space is a separable Hilbert space).

[F1]

Céa's estimate gives the Galerkin error at most $M / α$ times best approximation in $V$ (Céa quasi-optimality for coercive conforming Galerkin problems).

[F2]

The affine finite-element interpolant maps the declared regular zero-boundary class into $V_{h}$ and has $∥ w - I_{h} w ∥_{H^{1}} \leq C_{F E} h^{r} ∣ w ∣_{W^{r + 1, 2}}$ (Primal H¹ approximation on shape-regular affine finite-element meshes).

[F3]

Under the two exact dual regularity/approximation inequalities stated above and Countable Choice, the static Galerkin error has the conditional bound $∥ w - R_{h} w ∥_{L^{2}} \leq C_{A} h^{r + s} ∣ w ∣_{W^{r + 1, 2}}$ (The h^(r+s) L² rate under primal and dual approximation).

[F4]

The time-independent Ritz map is bounded linear; every $u \in H^{1} (0, T; V)$ has the unique continuous $V$ representative used for $u (0)$ , and its weak derivative commutes with $R_{h}$ (The elliptic Ritz projection for a conforming finite-element space, Existence, uniqueness, linearity, and boundedness of the Ritz projection, Time derivatives of fixed and differentiable comparison projections).

[F5]

The semidiscrete Ritz error satisfies the residual energy estimate; for this autonomous Ritz map its form residual vanishes (Semidiscrete Ritz error splitting and energy estimate).

Proof

1.1F1F2F6A1A3

For almost every $t$ , use the Ritz equation with target $w = u (t)$ . It is the conforming Galerkin solution for the bounded functional $F_{w} (v) = a (w, v)$ . By [F1] and [F2], $∥ R_{h} u (t) - u (t) ∥_{V} \leq \frac{M}{α} \inf_{v_{h} \in V_{h}} ∥ u (t) - v_{h} ∥_{V} \leq \frac{M}{α} C_{F E} h^{r} ∣ u (t) ∣_{W^{r + 1, 2}} .$ Taking the essential supremum proves the first estimate.

1.2F3F6A1A2A3

The target $w = u (t)$ satisfies the real scalar finite-element hypotheses of [F3]; its Ritz projection is its Galerkin solution for $F_{w}$ . The assumed uniform adjoint constants therefore give $∥ ρ (t) ∥_{H} \leq C_{A} h^{r + s} ∣ u (t) ∣_{W^{r + 1, 2}}$ . Apply the same argument to $w = u^{'} (t)$ for almost every $t$ .

2.1F4A1givenstep 1.2algebra

Since $R_{h}$ is fixed, [F4] gives $ρ^{'} = R_{h} u^{'} - u^{'}$ . The $w = u^{'}$ estimate in step 1.2 bounds $∥ ρ^{'} ∥_{L^{2} (H)}$ . The canonical embedding $H \to V_{h}^{*}$ has norm at most $c_{i}$ , hence $∥ ρ^{'} ∥_{L^{2} (V_{h}^{*})} \leq c_{i} C_{A} h^{r + s} ∥ u^{'} ∥_{L^{2} (W^{r + 1, 2})}$ .

3.1F5A1step 2.1given

For the error splitting $θ$ , the fixed Ritz orthogonality removes $a (t; ρ, v_{h})$ from the residual in [F5]. Its initial value is zero because $u_{h} (0) = R_{h} u (0)$ . Applying [F5] and step 2.1 gives $∥ θ (t) ∥_{H}^{2} + α ∥ θ ∥_{L^{2} (0, t; V)}^{2} \leq α^{- 1} c_{i}^{2} C_{A}^{2} h^{2 (r + s)} ∥ u^{'} ∥_{L^{2} (0, t; W^{r + 1, 2})}^{2} .$ Taking the supremum and square roots proves the two stated bounds for $θ$ .

4.1step 1.1step 3.1algebra∎

The decomposition $u_{h} - u = θ + ρ$ gives the $L^{\infty} (H)$ bound by the triangle inequality. For the $L^{2} (V)$ bound, step 1.1 gives $∥ ρ ∥_{L^{2} (V)} \leq \sqrt{T} (M / α) C_{F E} h^{r} ∥ u ∥_{L^{\infty} (W^{r + 1, 2})}$ , and step 3.1 bounds $θ$ ; adding them proves the displayed estimate. The $h^{r + 1}$ case follows by setting $s = 1$ in the conditional dual branch.

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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)