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  "body": "\n## Definition\n\nLet $d \\in \\set{2, 3}$ and let $\\Omega \\subset \\R^d$ be a bounded open set whose closure $\\overline{\\Omega}$ is a polygon ($d = 2$) or a polyhedron ($d = 3$).\nA *conforming mesh* of $\\overline{\\Omega}$ is a finite set $\\mathcal{T}$ of closed cells, each a simplex or each a convex quadrilateral or hexahedron, such that:\n\n1. the union of the cells of $\\mathcal{T}$ is $\\overline{\\Omega}$;\n2. the interiors of two distinct cells are disjoint;\n3. the intersection of two distinct cells is empty or a common face of both, that is, a common vertex, edge or facet.\n\n*Mesh generation* is the task of computing a conforming mesh of $\\overline{\\Omega}$ from a description of its boundary.\nUsually it must meet conditions on the size $h_K = \\operatorname{diam} K$ and the shape of each cell $K$.\nA typical shape condition is an upper bound on $h_K / \\rho_K$, where $\\rho_K$ is the radius of the largest ball contained in $K$.\n\n## Remarks\n\nWhen all cells are simplices, a conforming mesh is a [[triangulation|YH170WBQ6M]] of $\\overline{\\Omega}$, the underlying space of a [[simplicial complex|V8GZ0W1ZZ5]].\nExample: the unit square $[0,1]^2$ with the two triangles of vertices $(0,0), (1,0), (1,1)$ and $(0,0), (1,1), (0,1)$ is a conforming mesh.\nThe triangles cover the square, their interiors are disjoint, and they meet along the common edge from $(0,0)$ to $(1,1)$.\nNon-example: cover $[0,2] \\times [0,1]$ by the squares $[0,1] \\times [0,1]$, $[1,2] \\times [0, \\tfrac12]$ and $[1,2] \\times [\\tfrac12, 1]$.\nClauses 1 and 2 hold, but clause 3 fails: the first and second squares meet in the segment $\\set{1} \\times [0, \\tfrac12]$.\nThis segment is an edge of the second square but not of the first.\nThe point $(1, \\tfrac12)$ is called a hanging node.\nA family of meshes with $\\max_K h_K \\to 0$ and $h_K / \\rho_K$ bounded uniformly is called shape regular.\nFinite element error estimates are usually proved for such families.\n\n## Explanations\n\n### L1 human\n\nA mesh cuts a region into simple pieces that fit together edge to edge.\nThere are no gaps, no overlaps, and no corner of one piece in the middle of a side of another.\nMesh generation is the job of producing such a cutting, with pieces that are neither too big nor too thin.\n",
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