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A direct Strang-type energy error estimate for SIPDG

Statement

Assume Countable Choice and the mesh, coefficient, penalty, and exact-solution hypotheses of Exact consistency of the symmetric interior-penalty DG form and SIPDG coercivity above the discrete inverse-trace threshold. Let $u_{h} \in V_{h}^{b r}$ be the unique discrete solution, so $a_{h} (u_{h}, v_{h}) = ℓ_{h} (v_{h})$ for every $v_{h} \in V_{h}^{b r}$ . Let $C_{f l u x}$ be the constant in Discrete inverse-trace bound for SIPDG coefficient fluxes and put $C_{B} : = 2 + 2 γ^{- 1 / 2}$ . Then for every $w_{h} \in V_{h}^{b r}$ , $∥ u - u_{h} ∥_{D G} \leq (1 + 2 C_{B} \sqrt{1 + C_{f l u x}}) ∥ u - w_{h} ∥_{D G, *} .$ Consequently $∥ u - u_{h} ∥_{D G} \leq (1 + 2 C_{B} \sqrt{1 + C_{f l u x}}) \inf_{w_{h} \in V_{h}^{b r}} ∥ u - w_{h} ∥_{D G, *}$ . The augmented approximation term retains the face-flux trace needed to bound the SIP consistency terms; no approximation rate is asserted without a separate bound on that term.

Facts & Assumptions

Given: Countable Choice; the exact solution $u$ and discrete solution $u_{h}$ satisfying the equations in the Statement; the coercivity threshold; and the mesh-dependent expressions $∥ \cdot ∥_{D G}$ and $∥ \cdot ∥_{D G, *}$ .

[F1]

The form and energy expressions are defined on $D_{h}$ , which contains the cellwise $H^{2}$ exact solution and the broken polynomials. (Symmetric interior-penalty DG form and energy seminorm)

[F2]

For $γ \geq 4 C_{f l u x}$ , $a_{h} (v_{h}, v_{h}) \geq \frac{1}{2} ∥ v_{h} ∥_{D G}^{2}$ on $V_{h}^{b r}$ . (SIPDG coercivity above the discrete inverse-trace threshold)

[F3]

For the stated piecewise $H^{2}$ exact solution, $a_{h} (u, v_{h}) = ℓ_{h} (v_{h})$ for every $v_{h} \in V_{h}^{b r}$ . (Exact consistency of the symmetric interior-penalty DG form)

[F4]

For every $z_{h} \in V_{h}^{b r}$ , $\sum_{F} h_{F} ∥ q_{F} (z_{h}) ∥_{L^{2} (F)}^{2} \leq C_{f l u x} \sum_{K} \int_{K} \nabla z_{h}^{T} A_{K} \nabla z_{h}$ . (Discrete inverse-trace bound for SIPDG coefficient fluxes)

[F5]

The form obeys $∣ a_{h} (v, w) ∣ \leq C_{B} ∥ v ∥_{D G, *} ∥ w ∥_{D G, *}$ on $D_{h}$ , and $∥ \cdot ∥_{D G}$ is a seminorm there. (Boundedness of the SIPDG form in the augmented norm)

[F6]

Countable Choice is used by the inverse-trace and physical-simplex Green suppliers [F3]–[F4]. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

Proof

1.1F1F5given

The exact solution $u$ is cellwise $H^{2}$ and $u_{h}, w_{h}$ are broken polynomials, so $u - w_{h}$ and $u_{h} - w_{h}$ belong to $D_{h}$ . The boundedness constant for their form pairing is $C_{B} = 2 + 2 γ^{- 1 / 2}$ by [F5].

2.1F1F2F3F4F5F6step 1.1givenalgebra

Fix $w_{h}$ and set $z_{h} = u_{h} - w_{h}$ . If $∥ z_{h} ∥_{D G} = 0$ , then $∥ z_{h} ∥_{D G} \leq 2 C_{B} \sqrt{1 + C_{f l u x}} ∥ u - w_{h} ∥_{D G, *}$ is immediate. Otherwise, coercivity [F2], the discrete equation and exact consistency [F3] give $\frac{1}{2} ∥ z_{h} ∥_{D G}^{2} \leq a_{h} (z_{h}, z_{h}) = a_{h} (u - w_{h}, z_{h})$ . By [F4] and $\sum_{K} \int_{K} \nabla z_{h}^{T} A_{K} \nabla z_{h} \leq ∥ z_{h} ∥_{D G}^{2}$ , $∥ z_{h} ∥_{D G, *} \leq \sqrt{1 + C_{f l u x}} ∥ z_{h} ∥_{D G}$ . Apply [F5] and step 1.1 to obtain $\frac{1}{2} ∥ z_{h} ∥_{D G}^{2} \leq C_{B} \sqrt{1 + C_{f l u x}} ∥ u - w_{h} ∥_{D G, *} ∥ z_{h} ∥_{D G}$ . Dividing by $∥ z_{h} ∥_{D G} > 0$ proves $∥ z_{h} ∥_{D G} \leq 2 C_{B} \sqrt{1 + C_{f l u x}} ∥ u - w_{h} ∥_{D G, *}$ .

3.1F5step 2.1algebra∎

Since $∥ \cdot ∥_{D G}$ is a seminorm on $D_{h}$ by [F5], its triangle inequality and step 2.1 give $∥ u - u_{h} ∥_{D G} \leq ∥ u - w_{h} ∥_{D G} + ∥ w_{h} - u_{h} ∥_{D G} \leq (1 + 2 C_{B} \sqrt{1 + C_{f l u x}}) ∥ u - w_{h} ∥_{D G, *}$ . This holds for every $w_{h}$ , so taking the infimum proves both assertions.

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Sources

Jean-Luc Guermond, MATH 661 Fall 2025, Part VIII, Chapter 38: Discontinuous Galerkin (standard reference, not scraped)
Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn, and L. Donatella Marini, Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems, SIAM Journal on Numerical Analysis 39(5) (2002) (standard reference, not scraped)