Energy error estimate for the Crouzeix–Raviart Poisson method

The interpolant $I_h^{CR}u$ belongs to $V_h^{CR,0}$. Summing the local gradient estimates in [F3] over cells gives ${\sum }_K\mathrm{\parallel }\mathrm{\nabla }(u-I_K^{CR}u){\mathrm{\parallel }}_{L^2(K)}^2\le C{h}^{2r}{\sum }_K\mathrm{\mid }u{\mathrm{\mid }}_{H^{1+r}(K)}^2\le C{h}^{2r}\mathrm{\mid }u{\mathrm{\mid }}_{H^{1+r}(\mathrm{\Omega })}^2,$ since the cell-pair domains $K\times K$ are disjoint subsets of $\mathrm{\Omega }\times \mathrm{\Omega }$ for each gradient component.

The exact space $V=H_0^1(\mathrm{\Omega })$ is Hilbert in the gradient norm: it is Hilbert in $H^1$ by [F8], and continuous Poincare gives $\mathrm{\parallel }v{\mathrm{\parallel }}_{H^1}^2\le (1+C_P^2)\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_2^2$, so the equivalent gradient norm is complete. In that norm $a$ is bounded and coercive with constants $M=\alpha =1$. The discrete space $V_h$ has its explicit finite face basis [F7]; the common comparison inner product restricts to its norm by [F2], and [F9] makes $V_h$ Hilbert. Thus the two Hilbert-space hypotheses of [F6] hold. By [F10], the exact load satisfies $\mathrm{\mid }F(v)\mathrm{\mid }\le C_P\mathrm{\parallel }f{\mathrm{\parallel }}_2\mathrm{\parallel }v{\mathrm{\parallel }}_{\mathrm{♯},h}$, and the discrete load satisfies $\mathrm{\mid }{F}_h(w_h)\mathrm{\mid }\le C_D\mathrm{\parallel }f{\mathrm{\parallel }}_2\mathrm{\parallel }{\mathrm{\nabla }}_h{w}_h{\mathrm{\parallel }}_2\le C_D\mathrm{\parallel }f{\mathrm{\parallel }}_2\mathrm{\parallel }{w}_h{\mathrm{\parallel }}_{\mathrm{♯},h}$ using [F7]. Finally, Holder gives $\mathrm{\mid }{\tilde{a}}_h(z,w_h)\mathrm{\mid }\le \mathrm{\parallel }{\mathrm{\nabla }}_{h}z{\mathrm{\parallel }}_2\mathrm{\parallel }{\mathrm{\nabla }}_h{w}_h{\mathrm{\parallel }}_2\le \mathrm{\parallel }z{\mathrm{\parallel }}_{\mathrm{♯},h}\mathrm{\parallel }{w}_h{\mathrm{\parallel }}_{\mathrm{♯},h}$, so the bounded-extension constant is $M_h=1$.

By [F4], choose a constant $C_J$ such that $\mathrm{\parallel }{v}_h{\mathrm{\parallel }}_{\mathrm{♯},h}^2=\mathrm{\parallel }{\mathrm{\nabla }}_h{v}_h{\mathrm{\parallel }}_2^2+{\sum }_F{h}_F^{-1}\mathrm{\parallel }[v_h{]}_F{\mathrm{\parallel }}_2^2\le (1+C_J)\mathrm{\parallel }{\mathrm{\nabla }}_h{v}_h{\mathrm{\parallel }}_2^2=(1+C_J)a_h(v_h,v_h)\mathrm{.}$ Thus $a_h$ is coercive in the comparison norm with ${\alpha }_h=(1+C_J{)}^{-1}>0$, uniformly in the mesh.

If $V_h=\{0\}$, [F6] defines ${\delta }_h^{\mathrm{n}\mathrm{c}}(u)=0$, which satisfies the required bound. Otherwise, the residual in [F6] is ${\delta }_h^{\mathrm{n}\mathrm{c}}(u)={\sup }_{w_h\mathrm{\ne }0}\frac{\mathrm{\mid }{F}_h(w_h)-{\tilde{a}}_h(u,w_h)\mathrm{\mid }}{\mathrm{\parallel }{w}_h{\mathrm{\parallel }}_{\mathrm{♯},h}}\le C{h}^r\mathrm{\mid }u{\mathrm{\mid }}_{H^{1+r}(\mathrm{\Omega })},$ because [F5] gives the numerator bound and $\mathrm{\parallel }{w}_h{\mathrm{\parallel }}_{\mathrm{♯},h}\ge \mathrm{\parallel }{\mathrm{\nabla }}_h{w}_h{\mathrm{\parallel }}_2$.

On an interior facet, $[u-I_h^{CR}u]=-[I_h^{CR}u]$ because $u$ has a common trace; on a boundary facet its trace is $-I_h^{CR}u$ because $u$ has zero trace. The face terms in [F3], squared and summed over facets with each cell incident to at most $d+1$ facets, give ${\sum }_F{h}_F^{-1}\mathrm{\parallel }[u-I_h^{CR}u{]}_F{\mathrm{\parallel }}_{L^2(F)}^2\le C{h}^{2r}\mathrm{\mid }u{\mathrm{\mid }}_{H^{1+r}(\mathrm{\Omega })}^2\mathrm{.}$ Together with step 1.1, this proves $\mathrm{\parallel }u-I_h^{CR}u{\mathrm{\parallel }}_{\mathrm{♯},h}\le C{h}^r\mathrm{\mid }u{\mathrm{\mid }}_{H^{1+r}(\mathrm{\Omega })}$, hence the same upper bound for the infimum in [F6].

Apply [F6] using steps 2.1, 1.2, 1.3, and 1.4. The constants $M_h=1$ and ${\alpha }_h=(1+C_J{)}^{-1}$ are uniform, so its estimate gives $\mathrm{\parallel }u-u_h{\mathrm{\parallel }}_{\mathrm{♯},h}\le C{h}^r\mathrm{\mid }u{\mathrm{\mid }}_{H^{1+r}(\mathrm{\Omega })}$. The interpolation and consistency constants depend only on $d$, shape regularity, and $r$, and the jump constant $C_J$ has the same dependence. The Poincare constants $C_P,C_D$ in step 1.2 establish Hilbert-space/load hypotheses but do not enter the Strang error bound. Thus this proof yields an error constant depending only on $d$, shape regularity, and $r$ (and is compatible with the weaker dependence stated above). The broken-gradient term is one of the nonnegative terms in the squared comparison norm, yielding the final assertion.