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Discrete inverse-trace bound for SIPDG coefficient fluxes

Statement

Assume Countable Choice. Let $T$ , $A_{K}$ , $k$ , $λ$ , $Λ$ , $h_{F}$ , and $q_{F}$ be as in Symmetric interior-penalty DG form and energy seminorm. There is a constant $C_{f l u x} < \infty$ , depending only on $d$ , $k$ , $σ$ , $λ$ , and $Λ$ , such that every $v_{h} \in V_{h}^{b r}$ satisfies $\sum_{F \in F_{h}} h_{F} ∥ q_{F} (v_{h}) ∥_{L^{2} (F)}^{2} \leq C_{f l u x} \sum_{K \in T} \int_{K} A_{K} \nabla v_{h} \cdot \nabla v_{h} d x .$ The constant is uniform over the mesh family and does not depend on its cell sizes. In particular, the penalty threshold derived from this estimate is mesh-degree and shape dependent; it is not a universal numerical constant.

Facts & Assumptions

Given: Countable Choice; a shape-regular affine simplicial mesh with bound $σ$ ; $k \geq 1$ ; piecewise constant symmetric coefficients satisfying $λ I \leq A_{K} \leq Λ I$ ; and $v_{h} \in V_{h}^{b r}$ .

[F1]

The SIPDG definition gives $q_{F} (v_{h})$ as the arithmetic average of the adjacent coefficient fluxes, dotted with the fixed face normal; the boundary flux is one-sided. (Symmetric interior-penalty DG form and energy seminorm)

[F2]

For every cell polynomial and facet, the discrete trace estimate has scaling $∥ g ∥_{L^{2} (F)} \leq C h_{K}^{- 1 / 2} ∥ g ∥_{L^{2} (K)}$ ; constants depend on degree and shape regularity. (Polynomial inverse and discrete trace inequalities)

[F3]

The inverse-trace supplier explicitly assumes Countable Choice. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[A1]

Since $F \subset K$ and $h_{F} = diam (F)$ , $h_{F} \leq h_{K}$ for every incident cell.

Proof

1.1F1F2F3A1given

For each cell $K$ , $A_{K} \nabla (v_{h} ∣_{K})$ is a vector polynomial of degree at most $k - 1$ . Let $C_{i n v}$ be $\sqrt{d}$ times the trace constant in [F2] with polynomial degree $k - 1$ , so the estimate applies componentwise. Use $∣ A_{K} ξ ∣ \leq Λ ∣ ξ ∣$ . For an interior face, convexity of the squared norm gives $∣ {A \nabla_{h} v_{h}} ∣^{2} \leq \frac{1}{2} (∣ A_{K^{-}} \nabla v_{h}^{-} ∣^{2} + ∣ A_{K^{+}} \nabla v_{h}^{+} ∣^{2})$ ; for a boundary face there is one term. By [A1], $h_{F} \leq h_{K}$ , so $h_{F} ∥ q_{F} (v_{h}) ∥_{L^{2} (F)}^{2} \leq C_{i n v}^{2} Λ^{2} \sum_{K \in T_{F}} ∥ \nabla v_{h} ∥_{L^{2} (K)}^{2}$ , where $T_{F}$ is the set of one or two incident cells.

2.1step 1.1givenalgebra∎

Each simplex has $d + 1$ facets, so summing step 1.1 over all faces counts each cell at most $d + 1$ times. Ellipticity gives $\sum_{K} ∥ \nabla v_{h} ∥_{L^{2} (K)}^{2} \leq λ^{- 1} \sum_{K} \int_{K} A_{K} \nabla v_{h} \cdot \nabla v_{h}$ . Thus the required estimate holds with $C_{f l u x} : = (d + 1) C_{i n v}^{2} Λ^{2} / λ$ .

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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)