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  "body": "\n## Definition\n\nLet $\\mathcal{T}_0$ be an initial mesh of a domain $\\Omega$, and fix a discretization such as the [[finite element method|98GZ9DVHTA]].\nLet an error indicator assign a number $\\eta_K(\\mathcal{T}) \\ge 0$ to each element $K$ of each mesh $\\mathcal{T}$.\n*Adaptive mesh refinement* builds meshes $\\mathcal{T}_0, \\mathcal{T}_1, \\ldots$ by repeating four steps for $\\ell = 0, 1, \\ldots$:\n\n1. Solve: compute the discrete solution $u_\\ell$ on $\\mathcal{T}_\\ell$.\n2. Estimate: compute $\\eta_K = \\eta_K(\\mathcal{T}_\\ell)$ for all $K \\in \\mathcal{T}_\\ell$ from $u_\\ell$ and the data.\n3. Mark: choose a set $\\mathcal{M}_\\ell \\subseteq \\mathcal{T}_\\ell$ of elements with large indicators.\n4. Refine: subdivide at least the elements of $\\mathcal{M}_\\ell$, plus the neighbours needed to keep the mesh conforming, to get $\\mathcal{T}_{\\ell + 1}$.\n\nThe loop stops when $\\big(\\sum_K \\eta_K^2\\big)^{1/2}$ is below a tolerance.\n\n## Remarks\n\nA common marking rule is the Dörfler criterion with a parameter $0 \u003c \\theta \\le 1$.\nIt takes a set $\\mathcal{M}_\\ell$ of minimal size with $\\sum_{K \\in \\mathcal{M}_\\ell} \\eta_K^2 \\ge \\theta \\sum_{K \\in \\mathcal{T}_\\ell} \\eta_K^2$.\nFor the Poisson equation, a standard residual indicator combines $h_K^2 \\norm{f + \\Delta u_\\ell}_{L^2(K)}^2$ with the jumps of the normal derivative of $u_\\ell$ across the faces of $K$.\n\nExample: on the L-shaped domain $(-1, 1)^2 \\setminus [0, 1) \\times (-1, 0]$, the solution of $-\\Delta u = 0$ with boundary values $r^{2/3} \\sin(2\\varphi/3)$ is $u = r^{2/3} \\sin(2\\varphi/3)$ in polar coordinates $(r, \\varphi)$ with $0 \u003c \\varphi \u003c 3\\pi/2$.\nIts gradient blows up at the corner $0$, the indicators are largest there, and the meshes become graded towards $0$.\n\nNon-example: uniform refinement subdivides every element of $\\mathcal{T}_\\ell$ in each step.\nIts refinement does not depend on any indicator, so step 3 is missing.\n",
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