Symmetric interior-penalty discontinuous Galerkin

Assume Countable Choice. Let $d\ge 2$, let ${\widehat{S}}_d$ be the reference simplex, and let $F_K(\widehat{x})=b+B\widehat{x}$ map $\overline{{\widehat{S}}_d}$ affinely onto a nondegenerate closed simplex $K$, with $\det B>0$. Sobolev spaces on $K$ mean spaces on its open interior. For every scalar $u\in W^{1,2}(K)$ and vector field $G\in W^{1,2}(K;{\mathbb{R}}^d)$, each physical facet has bounded $L^2$ traces ${\gamma }_{F}u$ and ${\gamma }_{F}G$, obtained by affine transfer from the reference trace maps. These traces are independent of the chosen orientation-preserving reference map, and they satisfy ${\int }_{K}u\mathrm{div}G+\mathrm{\nabla }u\cdot Gdx={\sum }_{F\subset \mathrm{\partial }K}{\int }_F({\gamma }_{F}u)({\gamma }_{F}G)\cdot n_{K,F}dS,$ where $n_{K,F}$ is the outward unit normal.

Let $d\ge 2$ and let $\mathcal{T}$ be a finite matching affine simplicial mesh of a bounded polyhedral open set $\mathrm{\Omega }\subset {\mathbb{R}}^d$, with shape-regularity bound $\sigma$, as in Matching affine simplicial meshes and shape regularity. Write ${\mathcal{F}}_h^{\mathrm{i}\mathrm{n}\mathrm{t}}$ for its interior facets and ${\mathcal{F}}_h^{\mathrm{b}\mathrm{d}\mathrm{y}}$ for its boundary facets, and put ${\mathcal{F}}_h={\mathcal{F}}_h^{\mathrm{i}\mathrm{n}\mathrm{t}}\cup {\mathcal{F}}_h^{\mathrm{b}\mathrm{d}\mathrm{y}}$. Fix a degree $k\in {\mathbb{N}}_0$ for this broken space. The trial functions below belong to the broken scalar polynomial space $V_h^{\mathrm{b}\mathrm{r}}={\prod }_{K\in \mathcal{T}}{\mathbb{P}}_k(K;\mathbb{R})$ from Broken and conforming polynomial spaces on a matching mesh.

For each interior facet $F=K^{-}\cap K^{+}$, choose a unit normal $n_F$ pointing from $K^{-}$ to $K^{+}$. For a scalar or vector field $g$ with traces from both cells, define $[g{]}_F:=g^{-}-g^{+},\{g{\}}_F:=\frac{1}{2}(g^{-}+g^{+})\mathrm{.}$ For a boundary facet $F\subset \mathrm{\partial }K\cap \mathrm{\partial }\mathrm{\Omega }$, let $n_F=n_K$ be the outward unit normal and define $[g{]}_F:=g{\mathrm{\mid }}_K$ and $\{g{\}}_F:=g{\mathrm{\mid }}_K$. Reversing the chosen interior normal also reverses $[g{]}_F$, so the scalar jump/average contractions used in the form are independent of that choice.

For one-dimensional interval examples, use the following separate extension of the same formulas. At an interior point shared by a left cell $K^{-}$ and right cell $K^{+}$, orient from left to right and set $[g{]}_F=g^{-}-g^{+}$ and $\{g{\}}_F=(g^{-}+g^{+})\mathrm{/}2$. At either endpoint use the one-sided trace for both jump and average, and use counting measure on the zero-dimensional face. This extension is not an instance of the published $d\ge 2$ mesh definition above.

The following diagram records the interior orientation; the formulas above are the definitions, and the diagram is only a convention aid.

This is the scalar-jump convention: on an interior facet the corresponding vector jump is $[v{]}_F{n}_F$. On a boundary facet the jump is the trace itself; this is the convention that makes the penalty and consistency terms impose homogeneous Dirichlet data weakly. A normal-flux contraction with the scalar jump is unchanged when the interior orientation is reversed.

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 $w_h\in V_h^{\mathrm{b}\mathrm{r}}$.

Assume Countable Choice. Let $\mathcal{T}$, $A_K$, $k$, $\lambda$, $\mathrm{\Lambda }$, $h_F$, and $q_F$ be as in Symmetric interior-penalty DG form and energy seminorm. There is a constant $C_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}<\mathrm{\infty }$, depending only on $d$, $k$, $\sigma$, $\lambda$, and $\mathrm{\Lambda }$, such that every $v_h\in V_h^{\mathrm{b}\mathrm{r}}$ satisfies ${\sum }_{F\in {\mathcal{F}}_h}{h}_F\mathrm{\parallel }{q}_F(v_h){\mathrm{\parallel }}_{L^2(F)}^2\le C_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}{\sum }_{K\in \mathcal{T}}{\int }_K{A}_K\mathrm{\nabla }{v}_h\cdot \mathrm{\nabla }{v}_{h}dx\mathrm{.}$ 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.

Assume Countable Choice. Let $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{DG}$ 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 $\lambda >0$ and $\gamma >0$. Then $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{DG}$ is a norm on $V_h^{\mathrm{b}\mathrm{r}}$.

Assume Countable Choice and the SIPDG setting of Symmetric interior-penalty DG form and energy seminorm, with $\gamma >0$. For all $v,w\in {\mathcal{D}}_h$, $\mathrm{\mid }{a}_h(v,w)\mathrm{\mid }\le C_B\mathrm{\parallel }v{\mathrm{\parallel }}_{DG,\ast }\mathrm{\parallel }w{\mathrm{\parallel }}_{DG,\ast },C_B:=2+2{\gamma }^{-1\mathrm{/}2}\mathrm{.}$ The energy expression $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{DG}$ is also a seminorm on ${\mathcal{D}}_h$.

Assume Countable Choice. Let the mesh, coefficients, and SIPDG form be as in Symmetric interior-penalty DG form and energy seminorm. Suppose $u{\mathrm{\mid }}_K\in H^2(K)$ on every cell, its facet traces agree across every interior facet, its boundary trace is zero, and $f{\mathrm{\mid }}_K=-\mathrm{\nabla }\cdot (A_K\mathrm{\nabla }u)$ on each cell with $f\in L^2(\mathrm{\Omega })$. Assume also that the normal coefficient fluxes agree across interior facets: $(A_{K^{-}}\mathrm{\nabla }{u}^{-})\cdot n_F=(A_{K^{+}}\mathrm{\nabla }{u}^{+})\cdot n_F$. Then for every $w_h\in V_h^{\mathrm{b}\mathrm{r}}$, $a_h(u,w_h)={\mathrm{\ell }}_h(w_h):={\sum }_{K\in \mathcal{T}}{\int }_{K}f{w}_{h}dx\mathrm{.}$

Assume Countable Choice and the mesh and coefficient hypotheses of Symmetric interior-penalty DG form and energy seminorm. Let $e{\mathrm{\mid }}_K\in H^2(K)$ on every cell, and let $z{\mathrm{\mid }}_K\in H^2(K)$ be a dual solution with matching interior traces, zero boundary trace, matching normal coefficient fluxes, and $-\mathrm{\nabla }\cdot (A_K\mathrm{\nabla }z)=e$ on every cell. Then the SIPDG form, extended to these piecewise regular functions by its same formula, satisfies $a_h(e,z)=\mathrm{\parallel }e{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}^2\mathrm{.}$

Assume Countable Choice. Use the mesh, coefficient field, energy norm, and form of Symmetric interior-penalty DG form and energy seminorm, and let $C_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}$ be the constant of Discrete inverse-trace bound for SIPDG coefficient fluxes. If $\gamma \ge 4C_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}$, then $a_h(v_h,v_h)\ge \frac{1}{2}\mathrm{\parallel }{v}_h{\mathrm{\parallel }}_{DG}^2(v_h\in V_h^{\mathrm{b}\mathrm{r}})\mathrm{.}$ Thus the discrete bilinear form is coercive with constant $1\mathrm{/}2$ in the broken energy norm. For every $f\in L^2(\mathrm{\Omega })$, 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.

Assume Countable Choice and the SIPDG setting of Symmetric interior-penalty DG form and energy seminorm. Let $u$ satisfy all hypotheses of Exact consistency of the symmetric interior-penalty DG form, and let $u_h\in V_h^{\mathrm{b}\mathrm{r}}$ satisfy $a_h(u_h,v_h)={\mathrm{\ell }}_h(v_h)$ for all $v_h\in V_h^{\mathrm{b}\mathrm{r}}$. Set $e:=u-u_h$, let $V:=H_0^1(\mathrm{\Omega };\mathbb{R})=W_0^{1,2}(\mathrm{\Omega };\mathbb{R})$ as in Zero-boundary Sobolev space as a norm closure, and let $h:={\max }_{K\in \mathcal{T}}{h}_K$, with $0<h\le {\mathrm{\ell }}_{\mathrm{\Omega }}:=\mathrm{diam}(\mathrm{\Omega })$. Suppose $s\in (1\mathrm{/}2,1]$ and a dual solution $z_e\in V$ exists satisfying ${\sum }_{K\in \mathcal{T}}{\int }_K{A}_K\mathrm{\nabla }v\cdot \mathrm{\nabla }{z}_{e}dx={\int }_{\mathrm{\Omega }}vedx(v\in V),$ with the following properties:

1. Its one-sided normal coefficient-flux traces are in $L^2(F)$ in the weak Green sense defining ${\mathcal{D}}_h$. In the branch $s=1$, assume also $z_e{\mathrm{\mid }}_K\in H^2(K)$ on every cell, with matching interior value traces, zero boundary value trace, and matching interior normal coefficient fluxes; then SIPDG adjoint consistency for piecewise H2 dual solutions gives $a_h(e,z_e)=\mathrm{\parallel }e{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}^2$. In the branch $s<1$, assume this adjoint-consistency identity as an additional hypothesis. The weak dual equation gives $\mathrm{div}(A_K\mathrm{\nabla }{z}_e){\mathrm{\mid }}_K=-e{\mathrm{\mid }}_K\in L^2(K)$, so the stated flux and trace conditions place $z_e$ in the form domain ${\mathcal{D}}_h$ of Symmetric interior-penalty DG form and energy seminorm.
2. The specified normed dual-regularity space ${\mathcal{R}}_s$ is a subspace of $V\cap {\mathcal{D}}_h$, and $\mathrm{\parallel }{z}_e{\mathrm{\parallel }}_{{\mathcal{R}}_s}\le C_{\mathrm{r}\mathrm{e}\mathrm{g}}\mathrm{\parallel }e{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}$ for some $C_{\mathrm{r}\mathrm{e}\mathrm{g}}\ge 0$.
3. There is $z_h\in V_h^{\mathrm{b}\mathrm{r}}$ such that $\mathrm{\parallel }{z}_e-z_h{\mathrm{\parallel }}_{DG,\ast }\le C_{\mathrm{a}\mathrm{p}\mathrm{p}}{h}^s{\mathrm{\ell }}_{\mathrm{\Omega }}^{1-s}\mathrm{\parallel }{z}_e{\mathrm{\parallel }}_{{\mathcal{R}}_s}\mathrm{.}$ Here $C_{\mathrm{a}\mathrm{p}\mathrm{p}}\ge 0$.

Then $\mathrm{\parallel }e{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}\le (2+2{\gamma }^{-1\mathrm{/}2})C_{\mathrm{a}\mathrm{p}\mathrm{p}}{C}_{\mathrm{r}\mathrm{e}\mathrm{g}}{h}^s{\mathrm{\ell }}_{\mathrm{\Omega }}^{1-s}\mathrm{\parallel }e{\mathrm{\parallel }}_{DG,\ast }\mathrm{.}$ The theorem retains the dual regularity exponent and the half-order face-flux contribution in $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{DG,\ast }$. It does not infer the dual approximation estimate or adjoint-consistency identity from elliptic regularity alone.