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Boundedness of the SIPDG form in the augmented norm

Statement

Assume Countable Choice and the SIPDG setting of Symmetric interior-penalty DG form and energy seminorm, with $γ > 0$ . For all $v, w \in D_{h}$ , $∣ a_{h} (v, w) ∣ \leq C_{B} ∥ v ∥_{D G, *} ∥ w ∥_{D G, *}, C_{B} : = 2 + 2 γ^{- 1 / 2} .$ The energy expression $∥ \cdot ∥_{D G}$ is also a seminorm on $D_{h}$ .

Facts & Assumptions

Given: Countable Choice, the SIPDG form and augmented expression on $D_{h}$ , and $γ > 0$ .

[F1]

Symmetric interior-penalty DG form and energy seminorm: $a_{h}$ is the cell coefficient-gradient term, two symmetric flux-jump terms, and the penalty term; $∥ \cdot ∥_{D G, *}^{2}$ adds $\sum_{F} h_{F} ∥ q_{F} (\cdot) ∥_{L^{2} (F)}^{2}$ to the energy expression.

[F2]

Holder's inequality for integrals, including the endpoint cases: for $L^{2}$ functions, $∣ \int f g ∣ \leq ∥ f ∥_{2} ∥ g ∥_{2}$ .

[F3]

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

[F4]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is the declared Sobolev/trace background of the form definition [F1].

Proof

1.1F1F2F3F4givenalgebra

Set $X (v)^{2} : = \sum_{K} \int_{K} \nabla v^{T} A_{K} \nabla v d x$ and $J (v)^{2} : = \sum_{F} h_{F}^{- 1} ∥ [v]_{F} ∥_{L^{2} (F)}^{2}$ . For any real vectors $ξ, η$ , positivity of $A_{K}$ gives $(ξ + t η)^{T} A_{K} (ξ + t η) \geq 0$ for every real $t$ ; the discriminant of this quadratic implies $∣ ξ^{T} A_{K} η ∣ \leq (ξ^{T} A_{K} ξ)^{1 / 2} (η^{T} A_{K} η)^{1 / 2}$ . Applying [F2] on each cell and then [F3] to the finite cell sum gives $∣ \sum_{K} \int_{K} \nabla v^{T} A_{K} \nabla w ∣ \leq X (v) X (w)$ . The same two inequalities give $∣ γ \sum_{F} h_{F}^{- 1} \int_{F} [v]_{F} [w]_{F} ∣ \leq γ J (v) J (w)$ . Applying [F3] to the two-component vectors $(X (v), \sqrt{γ} J (v))$ and $(X (w), \sqrt{γ} J (w))$ shows that the absolute value of the sum of the volume and penalty terms is at most $∥ v ∥_{D G} ∥ w ∥_{D G} \leq ∥ v ∥_{D G, *} ∥ w ∥_{D G, *}$ .

2.1F1F2F3step 1.1algebra

For one flux term, weighted Cauchy–Schwarz using [F2] on each face and [F3] on the finite face sum gives $∣ \sum_{F} \int_{F} q_{F} (v) [w]_{F} ∣ \leq (\sum_{F} h_{F} ∥ q_{F} (v) ∥_{2}^{2})^{1 / 2} J (w) \leq γ^{- 1 / 2} ∥ v ∥_{D G, *} ∥ w ∥_{D G, *}$ . Interchanging $v, w$ gives the same bound for the other flux term. Adding these two bounds to step 1.1 proves the stated bilinear-form constant.

3.1F1step 1.1algebra∎

The energy pairing $b_{D G} (v, w) : = \sum_{K} \int_{K} \nabla v^{T} A_{K} \nabla w + γ \sum_{F} h_{F}^{- 1} \int_{F} [v]_{F} [w]_{F}$ is symmetric and nonnegative on the diagonal. Step 1.1 proves $∣ b_{D G} (v, w) ∣ \leq ∥ v ∥_{D G} ∥ w ∥_{D G}$ . Expanding $b_{D G} (v + w, v + w)$ and using this bound gives $∥ v + w ∥_{D G}^{2} \leq (∥ v ∥_{D G} + ∥ w ∥_{D G})^{2}$ ; the formula also gives nonnegativity and absolute homogeneity. Taking square roots proves the seminorm axioms.

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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)