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One-dimensional SUPG energy error bound with explicit tau dependence

Statement

Assume the homogeneous one-dimensional problem and uniform P1 SUPG form of One-dimensional stationary convection–diffusion model and uniform P1 space and One-dimensional residual SUPG form with the exact cell parameter. Assume Countable Choice, and let $u \in C^{2} ([0, 1])$ be the exact solution and $u_{h} \in V_{h}$ the unique discrete solution from One-dimensional P1 SUPG is consistent and coercive. For every continuous, piecewise-H1 function $w$ with $w (0) = w (1) = 0$ , define $∥ w ∥_{S D}^{2} : = ϵ ∥ w^{'} ∥_{L^{2} (0, 1)}^{2} + τ b^{2} ∥ w^{'} ∥_{L^{2} (0, 1)}^{2}, M : = ∥ u^{''} ∥_{L^{\infty} (0, 1)} .$ This is the product $L^{2}$ norm of $(\sqrt{ϵ} w^{'}, \sqrt{τ} ∣ b ∣ w^{'})$ , so it has the triangle inequality. Then $∥ u - u_{h} ∥_{S D} \leq [h \sqrt{ϵ + τ b^{2}} + h \sqrt{ϵ} + h ∣ b ∣ \sqrt{τ} + ϵ \sqrt{τ} + \frac{h^{2}}{\sqrt{τ}}] M .$ With $c_{*}$ from The one-dimensional SUPG parameter is positive and bounded, there is a universal $C_{*} > 0$ such that also $∥ u - u_{h} ∥_{S D} \leq C_{*} (h \sqrt{ϵ} + h^{3 / 2} \sqrt{∣ b ∣}) M .$

Facts & Assumptions

Given: Countable Choice; $ϵ > 0$ , $b \neq 0$ , a uniform mesh of size $h$ , a homogeneous exact solution $u \in C^{2} ([0, 1])$ , and its discrete SUPG solution $u_{h}$ .

[F1]

One-dimensional residual SUPG form with the exact cell parameter: for P1 functions $a_{h} (v_{h}, v_{h}) = ϵ ∥ v_{h}^{'} ∥_{2}^{2} + τ b^{2} ∥ v_{h}^{'} ∥_{2}^{2}$ , while for piecewise smooth $w$ its stabilization uses $- ϵ w^{''} + b w^{'}$ ; $τ = h (2 ∣ b ∣)^{- 1} (\coth (P e) - 1 / P e)$ .

[F2]

One-dimensional P1 SUPG is consistent and coercive: $a_{h} (u - u_{h}, v_{h}) = 0$ for every $v_{h} \in V_{h}$ , and $a_{h} (v_{h}, v_{h}) = ∥ v_{h} ∥_{S D}^{2}$ .

[F3]

One-dimensional P1 interpolation has a first-derivative error bound under Countable Choice: for the nodal P1 interpolant $I_{h} u$ , $∥ (u - I_{h} u)^{'} ∥_{2} \leq h ∥ u^{''} ∥_{\infty}$ .

[F4]

One-dimensional P1 interpolation has an L2 error bound: $∥ u - I_{h} u ∥_{2} \leq h^{2} ∥ u^{''} ∥_{\infty}$ .

[F5]

The one-dimensional SUPG parameter is positive and bounded: $τ < h / (2 ∣ b ∣)$ , $τ \geq (c_{*} / 2) \min {h^{2} / (2 ϵ), h / ∣ b ∣}$ , and $τ \leq h^{2} / (8 ϵ)$ .

[A1]

Countable Choice is assumed only for the imported interpolation result [F3] (The Axiom of Countable Choice ( ${A C}_{ω}$ )); no arbitrary choice is used elsewhere in this finite-mesh proof.

Proof

1.1A1F3F4givenconstruct

Let $I_{h} u$ be the nodal interpolant, $η : = u - I_{h} u$ , and $e_{h} : = u_{h} - I_{h} u$ . The homogeneous endpoint data imply $I_{h} u \in V_{h}$ , so $e_{h} \in V_{h}$ . By [F3] and [F4], $∥ η^{'} ∥_{2} \leq h M$ and $∥ η ∥_{2} \leq h^{2} M$ .

2.1F1F2step 1.1algebra

By [F2], $a_{h} (e_{h}, e_{h}) = a_{h} (η, e_{h})$ . The convection term satisfies $(b η^{'}, e_{h}) = - (b η, e_{h}^{'})$ : cell boundary terms cancel at interior nodes because $η$ and $e_{h}$ are continuous, and the two outer terms vanish. The diffusion and P1 stabilization terms are bounded by $\sqrt{ϵ} ∥ η^{'} ∥_{2} ∥ e_{h} ∥_{S D}$ and $∣ b ∣ \sqrt{τ} ∥ η^{'} ∥_{2} ∥ e_{h} ∥_{S D}$ . The additional consistency-extension term involving $- ϵ η^{''}$ is bounded by $ϵ \sqrt{τ} ∥ η^{''} ∥_{2} ∥ e_{h} ∥_{S D}$ , because $\sqrt{τ} ∣ b ∣ ∥ e_{h}^{'} ∥_{2} \leq ∥ e_{h} ∥_{S D}$ . The integrated convection term is bounded by $∥ η ∥_{2} ∥ e_{h} ∥_{S D} / \sqrt{τ}$ , since $\sqrt{τ} ∣ b ∣ ∥ e_{h}^{'} ∥_{2} \leq ∥ e_{h} ∥_{S D}$ . Therefore $∣ a_{h} (η, e_{h}) ∣ \leq (\sqrt{ϵ} ∥ η^{'} ∥_{2} + ∣ b ∣ \sqrt{τ} ∥ η^{'} ∥_{2} + ϵ \sqrt{τ} ∥ η^{''} ∥_{2} + \frac{∥ η ∥_{2}}{\sqrt{τ}}) ∥ e_{h} ∥_{S D}$ .

3.1F2step 2.1algebra

If $e_{h} = 0$ , its norm estimate is immediate. Otherwise, divide step 2.1 by $∥ e_{h} ∥_{S D}$ and use [F2] to get $∥ e_{h} ∥_{S D} \leq (\sqrt{ϵ} + ∣ b ∣ \sqrt{τ}) ∥ η^{'} ∥_{2} + ϵ \sqrt{τ} ∥ η^{''} ∥_{2} + ∥ η ∥_{2} / \sqrt{τ}$ .

4.1F1F3F4step 1.1step 3.1algebra

Since $I_{h} u$ is affine on each cell, $η^{''} = u^{''}$ cellwise and $∥ η^{''} ∥_{2} \leq M$ . With $u - u_{h} = η - e_{h}$ , the triangle inequality, [F1], [F3], [F4], and step 3.1 give $∥ u - u_{h} ∥_{S D} \leq [h \sqrt{ϵ + τ b^{2}} + h \sqrt{ϵ} + h ∣ b ∣ \sqrt{τ} + ϵ \sqrt{τ} + \frac{h^{2}}{\sqrt{τ}}] M$ .

5.1F1F5step 4.1algebra∎

From [F5], $h ∣ b ∣ \sqrt{τ} \leq h^{3 / 2} \sqrt{∣ b ∣ / 2}$ . If $P e \leq 1$ , then $τ \geq c_{*} h^{2} / (4 ϵ)$ , so $h^{2} / \sqrt{τ} \leq 2 c_{*}^{- 1 / 2} h \sqrt{ϵ}$ , and $ϵ \sqrt{τ} \leq h \sqrt{ϵ} / \sqrt{8}$ . If $P e \geq 1$ , then $τ \geq c_{*} h / (2 ∣ b ∣)$ , so $h^{2} / \sqrt{τ} \leq (2 / c_{*})^{1 / 2} h^{3 / 2} \sqrt{∣ b ∣}$ ; also $ϵ \leq ∣ b ∣ h / 2$ and $τ < h / (2 ∣ b ∣)$ imply $ϵ \sqrt{τ} \leq h^{3 / 2} \sqrt{∣ b ∣} / (2 \sqrt{2})$ . Finally $h \sqrt{ϵ + τ b^{2}} \leq h \sqrt{ϵ} + h ∣ b ∣ \sqrt{τ}$ . These inequalities prove the second estimate with a universal constant depending only on $c_{*}$ .

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Sources

Volker John, Numerical Methods for Convection-Dominated Problems, lecture notes, Chapter 5 (standard reference, not scraped)
Joachim Schöberl, Numerical Methods for Partial Differential Equations, Chapter 24, Finite element error analysis (standard reference, not scraped)