The Crouzeix–Raviart method for the homogeneous Poisson problem

Assume Countable Choice. Let $\mathrm{\Omega }$ and $\mathcal{T}$ be as in The global homogeneous Crouzeix–Raviart space, let $V:=H_0^1(\mathrm{\Omega })$, let $V_h:=V_h^{CR,0}$, and let $f\in L^2(\mathrm{\Omega })$. The exact Poisson form and load are $a(u,v):={\int }_{\mathrm{\Omega }}\mathrm{\nabla }u\cdot \mathrm{\nabla }vdx,F(v):={\int }_{\mathrm{\Omega }}fvdx(u,v\in V)\mathrm{.}$ An exact solution is a $u\in V$ satisfying $a(u,v)=F(v)$ for every $v\in V$. For $v_h,w_h\in V_h$, define the broken form and discrete load by $a_h(v_h,w_h):={\sum }_{K\in \mathcal{T}}{\int }_K\mathrm{\nabla }{v}_h\cdot \mathrm{\nabla }{w}_{h}dx,F_h(w_h):={\int }_{\mathrm{\Omega }}f{w}_{h}dx\mathrm{.}$ The discrete problem is to find $u_h\in V_h$ such that $a_h(u_h,w_h)=F_h(w_h)$ for every $w_h\in V_h$.

Put $X_h:=V+V_h$. For each cell facet $F\subset \mathrm{\partial }K$, let ${\gamma }_{F,K}$ denote the trace supplied by Fractional face traces and cellwise Green identity on a simplex. For $z\in X_h$ and $w_h\in V_h$, define the comparison-space extension ${\tilde{a}}_h(z,w_h):={\sum }_{K\in \mathcal{T}}{\int }_K\mathrm{\nabla }(z{\mathrm{\mid }}_K)\cdot \mathrm{\nabla }{w}_{h}dx\mathrm{.}$ Each restriction $z{\mathrm{\mid }}_K$ belongs to $H^1(K)$, and this extension agrees with $a_h$ when $z\in V_h$.

On an interior facet $F=K^{+}\cap K^{-}$ choose either ordering and define $[z{]}_F:={\gamma }_{F,K^{+}}z-{\gamma }_{F,K^{-}}z$; on a boundary facet of $K$ set $[z{]}_F:={\gamma }_{F,K}z$. The mesh-dependent comparison expression is $\mathrm{\parallel }z{\mathrm{\parallel }}_{\mathrm{♯},h}^2:={\sum }_{K\in \mathcal{T}}\mathrm{\parallel }\mathrm{\nabla }(z{\mathrm{\mid }}_K){\mathrm{\parallel }}_{L^2(K)}^2+{\sum }_{F\in {\mathcal{F}}^{\circ }\cup {\mathcal{F}}^{\mathrm{\partial }}}{h}_F^{-1}\mathrm{\parallel }[z{]}_F{\mathrm{\parallel }}_{L^2(F)}^2,z\in X_h\mathrm{.}$ The next items prove the required definiteness, stability, and consistency properties; this definition makes no assertion that $V_h\subset V$.