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Exact consistency of the symmetric interior-penalty DG form

Statement

Assume Countable Choice. Let the mesh, coefficients, and SIPDG form be as in Symmetric interior-penalty DG form and energy seminorm. Suppose $u ∣_{K} \in H^{2} (K)$ on every cell, its facet traces agree across every interior facet, its boundary trace is zero, and $f ∣_{K} = - \nabla \cdot (A_{K} \nabla u)$ on each cell with $f \in L^{2} (Ω)$ . Assume also that the normal coefficient fluxes agree across interior facets: $(A_{K^{-}} \nabla u^{-}) \cdot n_{F} = (A_{K^{+}} \nabla u^{+}) \cdot n_{F}$ . Then for every $w_{h} \in V_{h}^{b r}$ , $a_{h} (u, w_{h}) = ℓ_{h} (w_{h}) : = \sum_{K \in T} \int_{K} f w_{h} d x .$

Facts & Assumptions

Given: Countable Choice; a finite matching affine simplicial mesh; cellwise constant symmetric positive-definite $A_{K}$ ; a piecewise $H^{2}$ function $u$ with matching interior traces and zero boundary trace; a piecewise $L^{2}$ source $f = - \nabla \cdot (A_{K} \nabla u)$ ; and matching interior normal fluxes.

[F1]

The SIPDG form contains the broken volume term, the two symmetric flux-jump terms, and the jump penalty; the boundary jump equals the trace of its cell. (Symmetric interior-penalty DG form and energy seminorm, Jump and average conventions for symmetric interior-penalty DG)

[F2]

On each physical affine simplex, for scalar $v \in W^{1, 2}$ and vector field $G \in W^{1, 2}$ , the transferred face traces satisfy $\int_{K} v div G + \nabla v \cdot G = \sum_{F \subset \partial K} \int_{F} (γ_{F} v) (γ_{F} G) \cdot n_{K, F}$ . (Sobolev traces and Gauss–Green formula on physical simplices)

[F3]

Countable Choice is the stated hypothesis of the physical-simplex trace and Green identity [F2]. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F4]

$H^{2} (K)$ means $W^{2, 2} (K)$ and $W^{1, 2} (K)$ consists of $L^{2}$ classes with first weak derivatives in $L^{2}$ . (Integer-order Sobolev spaces and their norms)

Proof

1.1F1F2F3F4given

On each cell $K$ , take $v = w_{h} ∣_{K}$ and $G = A_{K} \nabla u$ . Since $u ∣_{K} \in H^{2} (K)$ , [F4] and constant $A_{K}$ give $G \in W^{1, 2} (K; R^{d})$ , so [F2] and the cell equation yield $\int_{K} A_{K} \nabla u \cdot \nabla w_{h} = \int_{K} f w_{h} + \sum_{F \subset \partial K} \int_{F} (A_{K} \nabla u \cdot n_{K, F}) w_{h} ∣_{K}$ .

2.1F1step 1.1givenalgebra

Sum step 1.1 over cells. On an interior facet the two outward normals are $n_{F}$ and $- n_{F}$ ; the assumed normal-flux equality makes the two boundary terms equal $\int_{F} q_{F} (u) [w_{h}]_{F}$ . On a boundary facet the single term is also $\int_{F} q_{F} (u) [w_{h}]_{F}$ by the one-sided convention. Therefore the broken volume term equals $ℓ_{h} (w_{h}) + \sum_{F} \int_{F} q_{F} (u) [w_{h}]_{F}$ .

3.1F1step 2.1givenalgebra∎

The matching traces of $u$ and its zero boundary trace give $[u]_{F} = 0$ on every facet. Substitution into the SIPDG form makes its second flux-jump term exactly cancel the face sum of step 2.1, while its adjoint-consistency term and penalty term vanish. Hence $a_{h} (u, w_{h}) = ℓ_{h} (w_{h})$ .

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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)