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The SIPDG broken energy expression is definite

Statement

Assume Countable Choice. Let $∥ \cdot ∥_{D G}$ be the energy expression from Symmetric interior-penalty DG form and energy seminorm on a finite matching simplicial mesh of a nonempty bounded polyhedral domain, with $λ > 0$ and $γ > 0$ . Then $∥ \cdot ∥_{D G}$ is a norm on $V_{h}^{b r}$ .

Facts & Assumptions

Given: Countable Choice; a finite face-to-face triangulation of a bounded polyhedral open set, $λ > 0$ , $γ > 0$ , and the broken polynomial space and energy expression of Symmetric interior-penalty DG form and energy seminorm.

[F1]

The squared energy expression is the sum of nonnegative cell-gradient terms and positive-weight squared jumps on every interior and boundary facet. (Symmetric interior-penalty DG form and energy seminorm)

[F2]

A matching mesh is a finite union of nondegenerate simplices with pairwise-disjoint interiors; each interior facet is shared and each boundary facet is exterior. (Matching affine simplicial meshes and shape regularity)

[F3]

Products of $L^{2}$ functions satisfy the integral Cauchy–Schwarz inequality. (Holder's inequality for integrals, including the endpoint cases)

[F4]

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)

[F5]

On each physical simplex, the Green identity applies to $u = 1$ and the vector field $G (x) = x$ , giving $d ∣ K ∣ = \sum_{F \subset \partial K} \int_{F} x \cdot n_{K, F} d S$ . (Sobolev traces and Gauss–Green formula on physical simplices)

[F6]

Countable Choice is the declared Sobolev/trace background of [F1] and the physical-simplex Green supplier [F5]. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

Proof

1.1F1givenalgebra

Suppose $∥ v_{h} ∥_{D G} = 0$ . Every cell-gradient term is zero, and positive definiteness of $A_{K}$ implies $\nabla (v_{h} ∣_{K}) = 0$ almost everywhere. Since $v_{h} ∣_{K}$ is a polynomial on the connected simplex $K$ , it is a constant $c_{K}$ .

2.1F1F2F5F6step 1.1algebra

Every interior-facet jump is zero by [F1], so adjacent cells have equal constants. Thus $c_{K}$ is constant on each connected component of the cell-facet adjacency graph. For any such component, sum [F5] over its finitely many cells. The cell volumes are positive, while the two oriented integrals on every shared facet cancel; if there were no unshared facet, this would give $d \sum_{K} ∣ K ∣ = 0$ , a contradiction. Every unshared facet is an exterior mesh facet by the matching property [F2]. On that facet the boundary jump is the cell trace itself, so [F1] forces the component's constant to be zero. Therefore $v_{h} = 0$ on every cell.

3.1F1F2F3F4step 2.1algebra∎

The expression is homogeneous under scalar multiplication. Define its associated pairing by $b_{D G} (v_{h}, w_{h}) : = \sum_{K} \int_{K} \nabla v_{h}^{T} A_{K} \nabla w_{h} + γ \sum_{F} h_{F}^{- 1} \int_{F} [v_{h}]_{F} [w_{h}]_{F}$ . For any vectors $ξ, η$ , the quadratic $(ξ + t η)^{T} A_{K} (ξ + t η)$ is nonnegative for all real $t$ ; if $η \neq 0$ , its discriminant is nonpositive, giving $∣ ξ^{T} A_{K} η ∣ \leq (ξ^{T} A_{K} ξ)^{1 / 2} (η^{T} A_{K} η)^{1 / 2}$ , and the case $η = 0$ is immediate. [F3] bounds the resulting cell integrals and each facet integral, while [F4] combines the finitely many cell and facet terms to give $∣ b_{D G} (v_{h}, w_{h}) ∣ \leq ∥ v_{h} ∥_{D G} ∥ w_{h} ∥_{D G}$ . Expanding $b_{D G} (v_{h} + w_{h}, v_{h} + w_{h})$ gives $∥ v_{h} + w_{h} ∥_{D G}^{2} \leq (∥ v_{h} ∥_{D G} + ∥ w_{h} ∥_{D G})^{2}$ . Thus the triangle inequality holds; together with step 2.1 this proves the norm axioms.

Depends on

Used by

SIPDG coercivity above the discrete inverse-trace threshold Theorem

Dependency tree · two levels

41 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)