Symmetric interior-penalty DG form and energy seminorm

Assume Countable Choice for the Sobolev notation below. Each cell $K$ is the closed simplex from Matching affine simplicial meshes and shape regularity; in $W^{m,2}(K)$ notation, $K$ means its open interior, and its facets are the boundary faces of that interior. Write $H^m(K):=W^{m,2}(K;\mathbb{R})$ for $m=1,2$. Use the mesh and jump convention of Jump and average conventions for symmetric interior-penalty DG. Specialize Broken and conforming polynomial spaces on a matching mesh to scalar real polynomials of a fixed degree $k\ge 1$, so $V_h^{\mathrm{b}\mathrm{r}}={\prod }_{K\in \mathcal{T}}{\mathbb{P}}_k(K;\mathbb{R})$. On every cell $K$, let $A_K\in {\mathbb{R}}^{d\times d}$ be a constant symmetric matrix. Assume common bounds $0<\lambda \le \mathrm{\Lambda }<\mathrm{\infty }$ such that $\lambda \mathrm{\mid }\xi {\mathrm{\mid }}^2\le {\xi }^T{A}_K\xi \le \mathrm{\Lambda }\mathrm{\mid }\xi {\mathrm{\mid }}^2(\xi \in {\mathbb{R}}^d,K\in \mathcal{T})\mathrm{.}$ Write $A{\mathrm{\mid }}_K=A_K$, let $h_F:=\mathrm{diam}(F)>0$, and fix a penalty parameter $\gamma >0$. Define ${\mathcal{D}}_h$ to consist of broken real-valued functions $v$ with $v{\mathrm{\mid }}_K\in H^1(K)$, distributional $\mathrm{div}(A_K\mathrm{\nabla }v{\mathrm{\mid }}_K)\in L^2(K)$, and an $L^2(F)$ one-sided normal coefficient-flux trace $r_{K,F}(v)$ on every facet $F$ of each cell $K$. Precisely, $r_{K,F}(v)$ is the unique $L^2(F)$ function satisfying ${\int }_K{A}_K\mathrm{\nabla }v\cdot \mathrm{\nabla }\varphi +\mathrm{div}(A_K\mathrm{\nabla }v)\varphi dx={\sum }_{F\subset \mathrm{\partial }K}{\int }_F{r}_{K,F}(v){\gamma }_F\varphi dS(\varphi \in C^{\mathrm{\infty }}(\bar{K}))\mathrm{.}$ Here ${\gamma }_F\varphi$ is the value trace from Sobolev traces and Gauss–Green formula on physical simplices; uniqueness follows by testing smooth functions supported near the relative interior of each facet, whose restrictions are dense in $L^2(F)$. If $F=K^{-}\cap K^{+}$ and $n_F$ points outward from $K^{-}$, set $q_F(v):=\frac{1}{2}(r_{K^{-},F}(v)-r_{K^{+},F}(v))$; on a boundary facet set $q_F(v):=r_{K,F}(v)$ with the outward normal. These definitions agree with $\{A{\mathrm{\nabla }}_{h}v{\}}_F\cdot n_F$ whenever the classical one-sided vector flux traces exist. Broken polynomials and cellwise $H^2$ functions belong to ${\mathcal{D}}_h$ by Sobolev traces and Gauss–Green formula on physical simplices. For $v,w\in {\mathcal{D}}_h$, define $a_h(v,w):={\sum }_{K\in \mathcal{T}}{\int }_K{A}_K\mathrm{\nabla }v\cdot \mathrm{\nabla }wdx-{\sum }_{F\in {\mathcal{F}}_h}{\int }_F{q}_F(v)[w{]}_{F}dS-{\sum }_{F\in {\mathcal{F}}_h}{\int }_F{q}_F(w)[v{]}_{F}dS+\gamma {\sum }_{F\in {\mathcal{F}}_h}{h}_F^{-1}{\int }_F[v{]}_F[w{]}_{F}dS,$ All facet integrals and $L^2(F)$ norms use $(d-1)$-dimensional surface measure for the stated mesh dimension $d\ge 2$. Counting measure is reserved for the separate one-dimensional interval examples in Jump and average conventions for symmetric interior-penalty DG. The expression is real and symmetric.

Define the energy expression and its augmented version on ${\mathcal{D}}_h$ by $\mathrm{\parallel }v{\mathrm{\parallel }}_{DG}^2:={\sum }_{K\in \mathcal{T}}{\int }_K{A}_K\mathrm{\nabla }v\cdot \mathrm{\nabla }vdx+\gamma {\sum }_{F\in {\mathcal{F}}_h}{h}_F^{-1}\mathrm{\parallel }[v{]}_F{\mathrm{\parallel }}_{L^2(F)}^2,$ $\mathrm{\parallel }v{\mathrm{\parallel }}_{DG,\ast }^2:=\mathrm{\parallel }v{\mathrm{\parallel }}_{DG}^2+{\sum }_{F\in {\mathcal{F}}_h}{h}_F\mathrm{\parallel }{q}_F(v){\mathrm{\parallel }}_{L^2(F)}^2\mathrm{.}$ The first expression is initially a seminorm on $V_h^{\mathrm{b}\mathrm{r}}$; its definiteness is not part of this definition. The stated Sobolev notation follows Integer-order Sobolev spaces and their norms. Cellwise $H^2$ functions have the required value and normal coefficient-flux traces by Sobolev traces and Gauss–Green formula on physical simplices, because $A_K$ is constant. The augmented expression is defined whenever $v\in {\mathcal{D}}_h$, so it applies in particular to the exact functions, dual functions and discrete polynomials used below.

For $f\in L^2(\mathrm{\Omega })$ define the discrete load ${\mathrm{\ell }}_h(w):={\sum }_{K\in \mathcal{T}}{\int }_{K}fwdx$. A symmetric interior-penalty DG solution is a function $u_h\in V_h^{\mathrm{b}\mathrm{r}}$ satisfying $a_h(u_h,w_h)={\mathrm{\ell }}_h(w_h)$ for every $w_h\in V_h^{\mathrm{b}\mathrm{r}}$.