Local approximation by the face-average Crouzeix–Raviart interpolant

On the reference simplex, each face mean of $v\in H^1$ is bounded by $C\mathrm{\parallel }v{\mathrm{\parallel }}_{H^1}$ by [F2]. The interpolant is a finite sum of these means times fixed affine shape polynomials, so it is bounded from $H^1$ to $H^1$. The trace-defined formula of The face-average Crouzeix–Raviart interpolant is therefore continuous on $H^{1+r}\subset H^1$.

First work on the reference simplex. Choose an affine polynomial $p$ with $\mathrm{\nabla }p={\mathrm{avg}}_K\mathrm{\nabla }u$ and ${\mathrm{avg}}_K(u-p)=0$. For $1\mathrm{/}2<r<1$, applying the fractional Poincare estimate proved in step 1.1 of Fractional face traces and cellwise Green identity on a simplex to each component of $\mathrm{\nabla }u$ gives $\mathrm{\parallel }\mathrm{\nabla }(u-p){\mathrm{\parallel }}_{L^2(K)}\le C\mathrm{\mid }u{\mathrm{\mid }}_{H^{1+r}(K)}$. The mean-zero Poincare–Wirtinger inequality [F3] then gives $\mathrm{\parallel }u-p{\mathrm{\parallel }}_{L^2(K)}\le C\mathrm{\parallel }\mathrm{\nabla }(u-p){\mathrm{\parallel }}_{L^2(K)}$. At $r=1$, apply [F3] to the mean-zero first derivatives, then again to $u-p$, obtaining $\mathrm{\parallel }u-p{\mathrm{\parallel }}_{H^1(K)}\le C\mathrm{\mid }u{\mathrm{\mid }}_{H^2(K)}$.

Since $p$ is affine, reproduction [F1] gives $u-I_K^{CR}u=(u-p)-I_K^{CR}(u-p)$. The reference stability from step 1.1 and the estimates of step 2.1 bound the reference $H^1$ error by $C\mathrm{\mid }u{\mathrm{\mid }}_{H^{1+r}}$. Under the affine map $F_K(\widehat{x})=b_K+A_K\widehat{x}$, each face surface Jacobian is constant, so face averages and the interpolant commute with pullback. The fractional gradient seminorm obeys $\mathrm{\mid }\widehat{u}{\mathrm{\mid }}_{H^{1+r}(\widehat{K})}\le C{h}_K^{1+r-d\mathrm{/}2}\mathrm{\mid }u{\mathrm{\mid }}_{H^{1+r}(K)}$: substitute $x=F_K(\widehat{x})$ in the double integral, use $\mathrm{\mid }{A}_{K}z\mathrm{\mid }\asymp h_K\mathrm{\mid }z\mathrm{\mid }$ and $\mathrm{\mid }\det A_K\mathrm{\mid }\asymp h_K^d$ from [F4], and include the factor $A_K^T$ in the pulled-back gradient. The physical gradient and $L^2$ error scaling factors are respectively $h_K^{d\mathrm{/}2-1}$ and $h_K^{d\mathrm{/}2}$, yielding the two stated powers.

Apply the scaled $H^1$ trace bound [F2] to $e=u-I_K^{CR}u$: $h_F^{-1\mathrm{/}2}\mathrm{\parallel }e{\mathrm{\parallel }}_{L^2(F)}\le C(h_K^{-1}\mathrm{\parallel }e{\mathrm{\parallel }}_{L^2(K)}+\mathrm{\parallel }\mathrm{\nabla }e{\mathrm{\parallel }}_{L^2(K)})\mathrm{.}$ Substituting step 3.1 gives the stated face estimate. On an interior facet, both adjacent restrictions of $u$ have the same Sobolev trace because $u\in H^1(\mathrm{\Omega })$; on a boundary facet the trace of $u\in H_0^1(\mathrm{\Omega })$ is zero. The global interpolant therefore has matching interior face means and zero boundary means.