Localized energy estimate for nested Galerkin solutions

Define an auxiliary coarse moment map $J:H_0^1(\mathrm{\Omega })\to V(\mathcal{T})$ for this nested pair. At each interior vertex $z$, choose the least-index incident coarse cell $L_z\mathrm{\notin }\mathcal{R}$ if one exists, otherwise the original selected cell $K_z$, and use its dual affine moment $q_z(w)={\int }_{L_z}w{\psi }_{z,L_z}$; at boundary vertices use zero. Set $Jw={\sum }_{z\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{r}}{q}_z(w){\phi }_z$. The same conforming zero-boundary membership argument as in [F2] applies. Its bounds are uniform: on a cell $K$ use the patch mean $c$ when $K$ misses the boundary and $c=0$ otherwise. Then $q_z(w)-c={\int }_{L_z}(w-c){\psi }_{z,L_z}$ for interior vertices, while boundary coefficient differences are zero. Because $L_z\subset {\omega }_K$ and $h_{L_z}\asymp h_K$, the scaled dual bound gives $\mathrm{\mid }{q}_z(w)-c\mathrm{\mid }\le C{h}_K^{-1}\mathrm{\parallel }w-c{\mathrm{\parallel }}_{L^2(L_z)}$. The hat bounds and patch Poincaré argument in the proof of [F2] therefore give $\mathrm{\parallel }w-Jw{\mathrm{\parallel }}_{L^2(K)}\le C{h}_K\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_{L^2({\omega }_K)}$ and $\mathrm{\parallel }\mathrm{\nabla }Jw{\mathrm{\parallel }}_{L^2(K)}\le C\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_{L^2({\omega }_K)}$. Applying the same scaled trace inequality as in [F2] gives its edge bound as well. All these operations have input $w\in H_0^1(\mathrm{\Omega })$; shifted functions enter only the local integrals.

Put $d=U_{\ast }-U$. The fine Galerkin equation and coarse equation imply $a(u-U_{\ast },d)=0$ and $a(u-U,Jd)=0$. Since $b$ is constant and $d\in H_0^1(\mathrm{\Omega })$, $\int (b\cdot \mathrm{\nabla }d)d=0$, so $a(d,d)=\mathrm{\parallel }\mathrm{\nabla }d{\mathrm{\parallel }}_2^2$. Also $a(u-U,d)=a(d,d)$ because $u-U=d+(u-U_{\ast })$ and fine orthogonality kills the latter term paired with $d$. Hence $\mathrm{\parallel }\mathrm{\nabla }d{\mathrm{\parallel }}_2^2=a(u-U,d-Jd)$.

If $K\mathrm{\notin }\mathcal{R}$, then $d$ is affine on $K$. At every interior vertex $z$ of $K$, the selected $L_z$ is also unchanged, so $d{\mathrm{\mid }}_{L_z}$ is affine and its dual moment equals the common nodal value $d(z)$; boundary coefficients equal $d(z)=0$. Consequently $Jd=d$ on every unchanged cell. The trace of $d-Jd$ also vanishes on every edge incident to an unchanged cell. Thus [F1] applied to step 2.1 contains only volume residuals on $\mathcal{R}$ and interior edges with both incident cells in $\mathcal{R}$. Use the volume and edge estimates proved in step 1.1. The surviving squared residual weights are bounded by $C{\sum }_{K\in \mathcal{R}}{\eta }_K(U{)}^2$, since each surviving edge contribution occurs in the indicators of its incident refined cells. Cauchy–Schwarz and uniformly bounded overlap of the coarse vertex patches bound the other factor by $C\mathrm{\parallel }\mathrm{\nabla }d{\mathrm{\parallel }}_2$, giving $\mathrm{\parallel }\mathrm{\nabla }d{\mathrm{\parallel }}_2^2\le C{({\sum }_{K\in \mathcal{R}}{\eta }_K(U{)}^2)}^{1\mathrm{/}2}\mathrm{\parallel }\mathrm{\nabla }d{\mathrm{\parallel }}_2$. If this gradient norm is zero the claim is immediate; otherwise divide by it and square.