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SIPDG coercivity above the discrete inverse-trace threshold

Statement

Assume Countable Choice. Use the mesh, coefficient field, energy norm, and form of Symmetric interior-penalty DG form and energy seminorm, and let $C_{f l u x}$ be the constant of Discrete inverse-trace bound for SIPDG coefficient fluxes. If $γ \geq 4 C_{f l u x}$ , then $a_{h} (v_{h}, v_{h}) \geq \frac{1}{2} ∥ v_{h} ∥_{D G}^{2} (v_{h} \in V_{h}^{b r}) .$ Thus the discrete bilinear form is coercive with constant $1 / 2$ in the broken energy norm. For every $f \in L^{2} (Ω)$ , the finite-dimensional SIPDG system has a unique solution. The sufficient threshold depends on the polynomial degree, shape regularity, and coefficient ellipticity bounds; no universal penalty value is asserted.

Facts & Assumptions

Given: Countable Choice; $k \geq 1$ ; symmetric cellwise constant $A_{K}$ with ellipticity bounds $0 < λ \leq Λ$ ; a finite matching shape-regular mesh; $γ \geq 4 C_{f l u x}$ ; and the SIPDG form and energy norm.

[F1]

The form is the cell energy minus the two symmetric flux-jump pairings plus $γ \sum_{F} h_{F}^{- 1} ([v]_{F}, [w]_{F})_{L^{2} (F)}$ . (Symmetric interior-penalty DG form and energy seminorm)

[F2]

The squared coefficient-flux trace satisfies $\sum_{F} h_{F} ∥ q_{F} (v_{h}) ∥_{L^{2} (F)}^{2} \leq C_{f l u x} \sum_{K} \int_{K} A_{K} \nabla v_{h} \cdot \nabla v_{h}$ . (Discrete inverse-trace bound for SIPDG coefficient fluxes)

[F3]

The broken energy expression is a norm on the finite-dimensional discontinuous polynomial space. (The SIPDG broken energy expression is definite)

[F4]

Countable Choice is the explicit hypothesis in the inverse-trace and energy-norm suppliers [F2]–[F3]. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F5]

The $L^{2}$ case of Holder's inequality gives $∣ \int_{F} f g d S ∣ \leq ∥ f ∥_{L^{2} (F)} ∥ g ∥_{L^{2} (F)}$ . (Holder's inequality for integrals, including the endpoint cases)

[F6]

Cauchy–Schwarz holds for every finite list of real numbers. (Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation)

Proof

1.1F1F2F4F5F6algebra

Put $X^{2} : = \sum_{K} \int_{K} A_{K} \nabla v_{h} \cdot \nabla v_{h}$ and $J^{2} : = \sum_{F} h_{F}^{- 1} ∥ [v_{h}]_{F} ∥_{L^{2} (F)}^{2}$ . The integral Cauchy–Schwarz inequality [F5] on each facet, followed by [F6] on the finite weighted face sum and [F2] give $∣ \sum_{F} \int_{F} q_{F} (v_{h}) [v_{h}]_{F} ∣ \leq (\sum_{F} h_{F} ∥ q_{F} (v_{h}) ∥_{F}^{2})^{1 / 2} J \leq \sqrt{C_{f l u x}} X J$ .

2.1F1step 1.1givenalgebra

By [F1], $a_{h} (v_{h}, v_{h}) = X^{2} - 2 \sum_{F} \int_{F} q_{F} (v_{h}) [v_{h}]_{F} + γ J^{2}$ . Young's inequality gives $2 \sqrt{C_{f l u x}} X J \leq \frac{1}{2} X^{2} + 2 C_{f l u x} J^{2}$ , hence $a_{h} (v_{h}, v_{h}) \geq \frac{1}{2} X^{2} + (γ - 2 C_{f l u x}) J^{2} \geq \frac{1}{2} (X^{2} + γ J^{2}) = \frac{1}{2} ∥ v_{h} ∥_{D G}^{2}$ .

3.1givenF1F3step 2.1algebra∎

The coefficient matrix of $a_{h}$ in any basis of the finite-dimensional space $V_{h}^{b r}$ is symmetric by [F1]. If its matrix kernel contained a nonzero vector, the corresponding $v_{h} \neq 0$ would satisfy $a_{h} (v_{h}, v_{h}) = 0$ , contradicting [F3] and step 2.1. Therefore the square matrix is injective, hence invertible in finite dimension, so every load has exactly one solution.

Depends on

Used by

A direct Strang-type energy error estimate for SIPDG Theorem

Dependency tree · two levels

39 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

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)