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SIPDG L2 error under adjoint consistency and dual approximation

Statement

Assume Countable Choice and the SIPDG setting of Symmetric interior-penalty DG form and energy seminorm. Let $u$ satisfy all hypotheses of Exact consistency of the symmetric interior-penalty DG form, and let $u_{h} \in V_{h}^{b r}$ satisfy $a_{h} (u_{h}, v_{h}) = ℓ_{h} (v_{h})$ for all $v_{h} \in V_{h}^{b r}$ . Set $e : = u - u_{h}$ , let $V : = H_{0}^{1} (Ω; R) = W_{0}^{1, 2} (Ω; R)$ as in Zero-boundary Sobolev space as a norm closure, and let $h : = \max_{K \in T} h_{K}$ , with $0 < h \leq ℓ_{Ω} : = diam (Ω)$ . Suppose $s \in (1 / 2, 1]$ and a dual solution $z_{e} \in V$ exists satisfying $\sum_{K \in T} \int_{K} A_{K} \nabla v \cdot \nabla z_{e} d x = \int_{Ω} v e d x (v \in V),$ with the following properties:

Its one-sided normal coefficient-flux traces are in $L^{2} (F)$ in the weak Green sense defining $D_{h}$ . In the branch $s = 1$ , assume also $z_{e} ∣_{K} \in H^{2} (K)$ on every cell, with matching interior value traces, zero boundary value trace, and matching interior normal coefficient fluxes; then SIPDG adjoint consistency for piecewise H2 dual solutions gives $a_{h} (e, z_{e}) = ∥ e ∥_{L^{2} (Ω)}^{2}$ . In the branch $s < 1$ , assume this adjoint-consistency identity as an additional hypothesis. The weak dual equation gives $div (A_{K} \nabla z_{e}) ∣_{K} = - e ∣_{K} \in L^{2} (K)$ , so the stated flux and trace conditions place $z_{e}$ in the form domain $D_{h}$ of Symmetric interior-penalty DG form and energy seminorm.
The specified normed dual-regularity space $R_{s}$ is a subspace of $V \cap D_{h}$ , and $∥ z_{e} ∥_{R_{s}} \leq C_{r e g} ∥ e ∥_{L^{2} (Ω)}$ for some $C_{r e g} \geq 0$ .
There is $z_{h} \in V_{h}^{b r}$ such that $∥ z_{e} - z_{h} ∥_{D G, *} \leq C_{a p p} h^{s} ℓ_{Ω}^{1 - s} ∥ z_{e} ∥_{R_{s}} .$ Here $C_{a p p} \geq 0$ .

Then $∥ e ∥_{L^{2} (Ω)} \leq (2 + 2 γ^{- 1 / 2}) C_{a p p} C_{r e g} h^{s} ℓ_{Ω}^{1 - s} ∥ e ∥_{D G, *} .$ The theorem retains the dual regularity exponent and the half-order face-flux contribution in $∥ \cdot ∥_{D G, *}$ . It does not infer the dual approximation estimate or adjoint-consistency identity from elliptic regularity alone.

Facts & Assumptions

Given: Countable Choice; the exact solution $u$ satisfying Exact consistency of the symmetric interior-penalty DG form and the discrete solution $u_{h}$ satisfying the equation in the Statement; the dual solution $z_{e} \in H_{0}^{1} (Ω; R)$ ; an exponent $s \in (1 / 2, 1]$ ; the branch-specific adjoint-consistency hypotheses in the Statement; the dual regularity and approximation inequalities in the Statement; and the augmented SIPDG expression.

[F1]

Exact consistency and the discrete equation imply $a_{h} (e, v_{h}) = 0$ for every $v_{h} \in V_{h}^{b r}$ . (Exact consistency of the symmetric interior-penalty DG form)

[F2]

On its form domain, the SIPDG bilinear form satisfies $∣ a_{h} (v, w) ∣ \leq C_{B} ∥ v ∥_{D G, *} ∥ w ∥_{D G, *}$ with $C_{B} = 2 + 2 γ^{- 1 / 2}$ . (Boundedness of the SIPDG form in the augmented norm)

[F3]

In the $s = 1$ branch, piecewise $H^{2}$ dual solutions with matching traces and normal flux satisfy $a_{h} (e, z_{e}) = ∥ e ∥_{L^{2} (Ω)}^{2}$ . (SIPDG adjoint consistency for piecewise H2 dual solutions)

[F4]

Countable Choice is the stated background for the exact-consistency, adjoint-consistency, physical-trace, and SIPDG form-domain suppliers used here. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F5]

The zero-boundary space $V$ is the norm closure of $C_{c}^{\infty} (Ω)$ in $W^{1, 2} (Ω)$ , so a smooth test compactly supported inside a cell belongs to $V$ . (Zero-boundary Sobolev space as a norm closure)

[F6]

The reference trace maps transfer to bounded value traces on each physical simplex facet for every cellwise $W^{1, 2}$ function. (Sobolev traces and Gauss–Green formula on physical simplices)

[F7]

The exact solution is cellwise $H^{2}$ by the hypotheses of exact consistency, and the SIPDG definition places cellwise $H^{2}$ functions and broken polynomials in $D_{h}$ . (Exact consistency of the symmetric interior-penalty DG form, Symmetric interior-penalty DG form and energy seminorm)

[A1]

For $s < 1$ , the dual solution is assumed to have the stated weak $L^{2}$ normal-flux traces and adjoint-consistency identity; this carrier does not prove the fractional trace theorem that would discharge those assumptions.

Proof

1.1F1F3F4A1givenchoose

The result is immediate if $∥ e ∥_{L^{2}} = 0$ . Otherwise choose $z_{h}$ from the stated approximation hypothesis. For $s = 1$ , [F3] gives $a_{h} (e, z_{e}) = ∥ e ∥_{L^{2}}^{2}$ under the stated piecewise- $H^{2}$ and trace-matching hypotheses; for $s < 1$ this identity is [A1]. Exact consistency plus the discrete equation give $a_{h} (e, z_{h}) = 0$ by [F1]. Hence $∥ e ∥_{L^{2}}^{2} = a_{h} (e, z_{e} - z_{h})$ .

1.2F2F4F5F6F7given

By [F7], $u$ is cellwise $H^{2}$ and $u_{h}, z_{h}$ are broken polynomials, so $e = u - u_{h}$ and $z_{h}$ belong to $D_{h}$ . For each cell $K$ , a test in $C_{c}^{\infty} (K)$ extended by zero belongs to $V$ by [F5]; the weak dual equation then gives $div (A_{K} \nabla z_{e}) ∣_{K} = - e ∣_{K} \in L^{2} (K)$ in distributions. Item 1 supplies the one-sided $L^{2}$ normal flux traces, and $z_{e} \in V$ supplies its cellwise $H^{1}$ restrictions and value traces by [F6]. Thus $z_{e} \in D_{h}$ and [F2] gives $∣ a_{h} (e, z_{e} - z_{h}) ∣ \leq C_{B} ∥ e ∥_{D G, *} ∥ z_{e} - z_{h} ∥_{D G, *}$ with $C_{B} = 2 + 2 γ^{- 1 / 2}$ .

2.1step 1.1step 1.2givenalgebra∎

Apply step 1.2 to step 1.1 and then use the dual approximation and regularity assumptions. This gives $∥ e ∥_{L^{2}}^{2} \leq C_{B} C_{a p p} C_{r e g} h^{s} ℓ_{Ω}^{1 - s} ∥ e ∥_{D G, *} ∥ e ∥_{L^{2}}$ . Divide by the positive $∥ e ∥_{L^{2}}$ to obtain the asserted estimate.

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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)