Residual indicators and data oscillation for P1 AFEM

Assume Countable Choice for the $L^2$ and Sobolev conventions of The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$).

All functions and data are real-valued. Let $\mathrm{\Omega }\subset {\mathbb{R}}^2$ be a bounded polygon and let $\mathcal{T}$ be a finite conforming matching triangulation by nondegenerate affine triangles, with maximum shape-regularity constant $\mathrm{\Gamma }$. Write $h_K=\mathrm{diam}K$ and $h_E=\mathrm{diam}E$. Let $V(\mathcal{T})$ be the continuous, piecewise-affine zero-boundary subspace of Broken and conforming polynomial spaces on a matching mesh, which lies in $H_0^1(\mathrm{\Omega };\mathbb{R})$ by Continuous matching piecewise polynomials belong to W1p and Zero-boundary conforming polynomials lie in W0 1p and satisfy Poincare. The exact weak model is $-\mathrm{\Delta }u+b\cdot \mathrm{\nabla }u=f,u\in H_0^1(\mathrm{\Omega };\mathbb{R}),f\in L^2(\mathrm{\Omega };\mathbb{R}),b\in {\mathbb{R}}^2\text{ constant}\mathrm{.}$ The weak form is ${\int }_{\mathrm{\Omega }}\mathrm{\nabla }u\cdot \mathrm{\nabla }v+(b\cdot \mathrm{\nabla }u)vdx={\int }_{\mathrm{\Omega }}fvdx$ for every $v\in H_0^1(\mathrm{\Omega };\mathbb{R})$. The case $b=0$ is the symmetric Laplace model. Given $U\in V(\mathcal{T})$, its cell residual is $R_K(U)=f{\mathrm{\mid }}_K-b\cdot \mathrm{\nabla }(U{\mathrm{\mid }}_K)$; this is $f+\mathrm{\Delta }U-b\cdot \mathrm{\nabla }U$ because $\mathrm{\Delta }U=0$ on each cell. For an interior edge $E=K^{+}\cap K^{-}$ define the outward flux jump $J_E(U)=\mathrm{\nabla }U{\mathrm{\mid }}_{K^{+}}\cdot n_{K^{+}}+\mathrm{\nabla }U{\mathrm{\mid }}_{K^{-}}\cdot n_{K^{-}},$ where $n_{K^{\pm }}$ are the unit outward normals. Set ${\eta }_K(U{)}^2=h_K^2\mathrm{\parallel }{R}_K(U){\mathrm{\parallel }}_{L^2(K)}^2+\frac{1}{2}{\sum }_{E\subset \mathrm{\partial }K,E\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{r}}{h}_E\mathrm{\parallel }{J}_E(U){\mathrm{\parallel }}_{L^2(E)}^2,$ and $\eta (U,\mathcal{S}{)}^2={\sum }_{K\in \mathcal{S}}{\eta }_K(U{)}^2$ for $\mathcal{S}\subset \mathcal{T}$. The factor $1\mathrm{/}2$ assigns each interior-edge contribution once globally. Let ${\mathrm{\Pi }}_K^0$ be the $L^2(K)$ projection onto constants. The cell data oscillation is ${\mathrm{osc}}_K(U{)}^2=h_K^2\mathrm{\parallel }{R}_K(U)-{\mathrm{\Pi }}_K^0{R}_K(U){\mathrm{\parallel }}_{L^2(K)}^2,\mathrm{osc}(U,\mathcal{S}{)}^2={\sum }_{K\in \mathcal{S}}{\mathrm{osc}}_K(U{)}^2\mathrm{.}$ Since $b\cdot \mathrm{\nabla }U$ is constant on $K$, this oscillation equals $h_K^2\mathrm{\parallel }f-{\mathrm{\Pi }}_K^{0}f{\mathrm{\parallel }}_2^2$.

For a separate definitional interface, consider the symmetric diffusion model $-\mathrm{div}(A\mathrm{\nabla }u)=f$, where $A$ is a symmetric uniformly elliptic matrix field in the sense of Uniformly elliptic divergence-form operators and their sesquilinear forms and is Lipschitz on each cell. Its cell residual for a piecewise-affine $U$ is $R_K^A(U)=f+\mathrm{div}(A\mathrm{\nabla }U)$, and its outward conormal-flux jump is the sum of $A\mathrm{\nabla }U\cdot n_K$ from the two incident cells. Let ${\mathrm{\Pi }}_E^0$ be the $L^2(E)$ projection onto constants. The projected edge term $h_E\mathrm{\parallel }{J}_E^A-{\mathrm{\Pi }}_E^0{J}_E^A{\mathrm{\parallel }}_{L^2(E)}^2$ records nonconstant conormal flux. The reliability, contraction, and complexity results authored here concern $A=I$ only; this extension supplies notation and no coefficient-dependent theorem.

The matching affine mesh and continuous broken-polynomial conventions are those of Matching affine simplicial meshes and shape regularity and Broken and conforming polynomial spaces on a matching mesh. The proof of the weak representation uses Sobolev face traces and Gauss–Green on a reference simplex and Interior facet orientations cancel in a matching mesh.