Residual reliability and local efficiency for P1 AFEM

Put $e=u-U$ and write $a(w,v)={\int }_{\mathrm{\Omega }}\mathrm{\nabla }w\cdot \mathrm{\nabla }v+(b\cdot \mathrm{\nabla }w)v$. By the exact Galerkin equation, $a(U,v_h)={\int }_{\mathrm{\Omega }}f{v}_h$ for every $v_h\in V(\mathcal{T})$; subtracting from the weak equation gives $a(e,v_h)=0$. For $w\in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$, constancy of $b$ gives $\int (b\cdot \mathrm{\nabla }w)w=\frac{1}{2}\int b\cdot \mathrm{\nabla }(w^2)=0$. The map $w\mapsto \int (b\cdot \mathrm{\nabla }w)w$ is continuous in $H^1$, so density extends this identity to all $w\in H_0^1$. Hence $a(e,e)=\mathrm{\parallel }\mathrm{\nabla }e{\mathrm{\parallel }}_2^2$ and, since $I_{\mathcal{T}}e\in V(\mathcal{T})$, $\mathrm{\parallel }\mathrm{\nabla }e{\mathrm{\parallel }}_2^2=a(e,e-I_{\mathcal{T}}e)$. Apply [F1] to $e-I_{\mathcal{T}}e$. Cauchy–Schwarz, the cell and edge interpolation bounds [F2], and finite overlap of vertex patches give $\mathrm{\parallel }\mathrm{\nabla }e{\mathrm{\parallel }}_2^2\le C\eta (U,\mathcal{T})\mathrm{\parallel }\mathrm{\nabla }e{\mathrm{\parallel }}_2\mathrm{.}$ If $\mathrm{\parallel }\mathrm{\nabla }e{\mathrm{\parallel }}_2=0$ reliability is immediate; otherwise divide by it and square. This proves the global upper bound.

For each cell let ${\bar{R}}_K={\mathrm{\Pi }}_K^0{R}_K$, and let ${\lambda }_1,{\lambda }_2,{\lambda }_3$ be its barycentric coordinates. The polynomial ${\psi }_K=27{\lambda }_1{\lambda }_2{\lambda }_3$, extended by zero outside $K$, has zero trace on $\mathrm{\partial }K$, satisfies $0\le {\psi }_K\le 1$, and has ${\int }_K{\psi }_K=(9\mathrm{/}20)\mathrm{\mid }K\mathrm{\mid }$. Thus $v_K={\bar{R}}_K{\psi }_K$ is an admissible $H_0^1$ test by [F5]. By affine scaling and [F3], $\mathrm{\parallel }{v}_K{\mathrm{\parallel }}_{L^2(K)}\le C\mathrm{\parallel }{\bar{R}}_K{\mathrm{\parallel }}_{L^2(K)}$ and $\mathrm{\parallel }\mathrm{\nabla }{v}_K{\mathrm{\parallel }}_{L^2(K)}\le C{h}_K^{-1}\mathrm{\parallel }{\bar{R}}_K{\mathrm{\parallel }}_{L^2(K)}$. The residual identity has no edge term for this test, so ${\bar{R}}_K{\int }_K{v}_K=a(e,v_K)+{\int }_K({\bar{R}}_K-R_K(U))v_K$. The left side is at least $c\mathrm{\parallel }{\bar{R}}_K{\mathrm{\parallel }}_{L^2(K)}^2$, while the two terms on the right are bounded by $C(1+\mathrm{\mid }b\mathrm{\mid }{h}_K)h_K^{-1}\mathrm{\parallel }\mathrm{\nabla }e{\mathrm{\parallel }}_{L^2(K)}\mathrm{\parallel }{\bar{R}}_K{\mathrm{\parallel }}_{L^2(K)}$ and $C\mathrm{\parallel }{R}_K-{\bar{R}}_K{\mathrm{\parallel }}_{L^2(K)}\mathrm{\parallel }{\bar{R}}_K{\mathrm{\parallel }}_{L^2(K)}$, respectively. If ${\bar{R}}_K\mathrm{\ne }0$, division and multiplication by $h_K$ give $h_K\mathrm{\parallel }{\bar{R}}_K{\mathrm{\parallel }}_{L^2(K)}\le C(1+\mathrm{\mid }b\mathrm{\mid }{h}_K)\mathrm{\parallel }\mathrm{\nabla }e{\mathrm{\parallel }}_{L^2(K)}+C{h}_K\mathrm{\parallel }{R}_K-{\bar{R}}_K{\mathrm{\parallel }}_{L^2(K)}$; if it is zero, the same bound for $h_K\mathrm{\parallel }{R}_K{\mathrm{\parallel }}_{L^2(K)}$ follows from $R_K=R_K-{\bar{R}}_K$. Hence in all cases $h_K\mathrm{\parallel }{R}_K{\mathrm{\parallel }}_{L^2(K)}\le C(1+\mathrm{\mid }b\mathrm{\mid }{h}_K)\mathrm{\parallel }\mathrm{\nabla }e{\mathrm{\parallel }}_{L^2(K)}+C{\mathrm{osc}}_K(U)$.

On an interior edge $E=K^{+}\cap K^{-}$ write ${\omega }_E=K^{+}\cup K^{-}$, and let ${\lambda }_i^{\pm },{\lambda }_j^{\pm }$ be the barycentric coordinates of its endpoints on the two incident cells. Define ${\psi }_E=4{\lambda }_i^{+}{\lambda }_j^{+}$ on $K^{+}$, ${\psi }_E=4{\lambda }_i^{-}{\lambda }_j^{-}$ on $K^{-}$, and zero outside their union. These traces agree on $E$, vanish on the other edges of the patch, and satisfy ${\int }_E{\psi }_E=(2\mathrm{/}3)h_E$, $\mathrm{\parallel }{\psi }_E{\mathrm{\parallel }}_{L^2(K^{+}\cup K^{-})}\le C{h}_E$, and $\mathrm{\parallel }\mathrm{\nabla }{\psi }_E{\mathrm{\parallel }}_{L^2(K^{+}\cup K^{-})}\le C$ by [F3]. The function ${\psi }_E$ is an admissible $H_0^1$ test by [F5]. Put $J=J_E(U)$, which is constant for P1 with $A=I$, and test [F1] with $v_E=J{\psi }_E$. All edge terms other than the one on $E$ vanish. Its magnitude is $(2\mathrm{/}3)h_E{J}^2$. The cell terms are bounded by $C{h}_E\mathrm{\mid }J\mathrm{\mid }(\mathrm{\parallel }{R}_{K^{+}}(U){\mathrm{\parallel }}_2+\mathrm{\parallel }{R}_{K^{-}}(U){\mathrm{\parallel }}_2)$, and the form term by $C(1+\mathrm{\mid }b\mathrm{\mid }{h}_E)\mathrm{\mid }J\mathrm{\mid }\mathrm{\parallel }\mathrm{\nabla }e{\mathrm{\parallel }}_{L^2(K^{+}\cup K^{-})}$. Use step 1.2, $h_E\asymp h_{K^{\pm }}$, and the definitions of the oscillations to obtain $h_E\mathrm{\mid }J\mathrm{\mid }\le C(1+\mathrm{\mid }b\mathrm{\mid }{h}_{\max })(\mathrm{\parallel }\mathrm{\nabla }e{\mathrm{\parallel }}_{L^2(K^{+}\cup K^{-})}+\mathrm{osc}(U,K^{+}\cup K^{-}))$; the bound is trivial if $J=0$. Since $h_E^{1\mathrm{/}2}\mathrm{\parallel }{J}_E(U){\mathrm{\parallel }}_{L^2(E)}=h_E\mathrm{\mid }J\mathrm{\mid }$, summing with bounded patch overlap and the cell estimate of step 1.2 proves the local efficiency assertion.

Set $d=U_{\ast }-U$. For every original cell $L$ in $\omega (M)$, use its fixed uniform-three-generation pattern from [F4]. Its P1 hat ${\varphi }_L$ at the interior median midpoint is zero on $\mathrm{\partial }L$. Exactly four of the eight equal-area leaves meet that midpoint, so ${\int }_L{\varphi }_L=4(\mathrm{\mid }L\mathrm{\mid }\mathrm{/}8)\mathrm{/}3=\mathrm{\mid }L\mathrm{\mid }\mathrm{/}6$. Reference scaling gives $\mathrm{\parallel }{\varphi }_L{\mathrm{\parallel }}_{L^2(L)}\le C{h}_L$ and $\mathrm{\parallel }\mathrm{\nabla }{\varphi }_L{\mathrm{\parallel }}_{L^2(L)}\le C$. Extend by zero and prolong to ${\mathcal{T}}_{\ast }$: containment of the fixed pattern makes this a fine P1 function even after arbitrary additional closure or refinement. For every fine test $v_{\ast }$, fine Galerkin orthogonality gives $a(u-U,v_{\ast })=a(d,v_{\ast })$. Testing with $v_{\ast }={\bar{R}}_L{\varphi }_L$, where ${\bar{R}}_L={\mathrm{\Pi }}_L^0{R}_L$, yields the cell estimate by exactly the residual splitting and zero-mean remainder argument of step 1.2, now with ${\int }_L{\varphi }_L=\mathrm{\mid }L\mathrm{\mid }\mathrm{/}6$: $h_L\mathrm{\parallel }{R}_L(U){\mathrm{\parallel }}_{L^2(L)}\le C(1+\mathrm{\mid }b\mathrm{\mid }{h}_{\max })\mathrm{\parallel }\mathrm{\nabla }d{\mathrm{\parallel }}_{L^2(L)}+C{\mathrm{osc}}_L(U)\mathrm{.}$ The case ${\bar{R}}_L=0$ uses only the oscillation remainder, so no zero quantity is divided by. This bound is now available on every neighbor needed below.

For each interior edge $E$ incident to a cell of $M$, both original incident cells belong to $\omega (M)$. Define a common trace tent $t_E$ on $E$: it is zero at its two endpoints, one at its midpoint, and affine on each half. On each incident cell's fixed three-generation pattern, prescribe nodal values $t_E(z)$ at nodes on $E$ and zero at every pattern node off $E$, then interpolate P1. In particular any quarter points on a reference edge have value $1\mathrm{/}2$, rather than zero. The traces from the two cells equal the same full tent even when their reference labels differ; all other original edge traces are zero. Thus the two pieces glue to a continuous function ${\psi }_E$ supported on ${\omega }_E$, zero on its boundary, with ${\int }_E{\psi }_E=h_E\mathrm{/}2,\mathrm{\parallel }{\psi }_E{\mathrm{\parallel }}_{L^2({\omega }_E)}\le C{h}_E,\mathrm{\parallel }\mathrm{\nabla }{\psi }_E{\mathrm{\parallel }}_{L^2({\omega }_E)}\le C\mathrm{.}$ Indeed each cell uses eight fixed-shape leaves, nodal values in $[0,1]$, and affine scaling; shape regularity gives $h_L\asymp h_E$ for the two cells. Prolong these reference functions to ${\mathcal{T}}_{\ast }$, retaining their values, support, integrals and bounds; do not replace them by nodal hats on the possibly much finer actual mesh. For $J=J_E(U)$, test with $J{\psi }_E$. The only original edge term is $J^2{h}_E\mathrm{/}2$, so the residual identity and local form bound give $h_E\mathrm{\mid }J\mathrm{\mid }\le C(1+\mathrm{\mid }b\mathrm{\mid }{h}_E)\mathrm{\parallel }\mathrm{\nabla }d{\mathrm{\parallel }}_{L^2({\omega }_E)}+C{h}_E{\sum }_{L\subset {\omega }_E}\mathrm{\parallel }{R}_L(U){\mathrm{\parallel }}_{L^2(L)}\mathrm{.}$ For $J=0$ this is immediate. Applying step 2.2 on both incident cells yields the desired edge estimate in terms of correction energy and oscillation on ${\omega }_E$. Summing squared cell and edge estimates over $M$ has bounded overlap: each triangle belongs to at most its own and its three neighbors' patches. This proves the marked estimate with precisely $\omega (M)$ on the right and constants independent of all extra refinement.