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  "body": "\n## Definition\n\nLet $\\mathcal{T}_h$ be a mesh of a domain $\\Omega \\subset \\R^d$ with nodes $x_i$, $i \\in I$.\nLet $N_i$ be standard [[finite element|98GZ9DVHTA]] shape functions with $\\sum_{i \\in I} N_i = 1$ on $\\Omega$, a [[partition of unity|NH739XXVXK]].\nLet $\\psi \\colon \\Omega \\to \\R$ be an enrichment function that carries a known local feature of the solution.\nLet $J \\subseteq I$ be the set of nodes whose shape function support meets the region of that feature.\nThe *extended finite element method* is the [[Galerkin method|Y5B0PEBKE8]] in the space of functions\n$$u_h(x) = \\sum_{i \\in I} N_i(x) \\, u_i + \\sum_{j \\in J} N_j(x) \\, \\psi(x) \\, a_j,$$\nwith unknown coefficients $u_i$ and $a_j$.\n\n## Remarks\n\nThe extra functions $N_j \\psi$ have small support, so the system matrix stays sparse.\nIf all $a_j$ equal one value $a$, then on a region where $\\sum_{j \\in J} N_j = 1$ the enrichment equals $a \\psi$ exactly.\nCommon choices of $\\psi$ are a [[Heaviside function|09C4JP7Y0M]] across a crack, and the singular functions of the crack tip, which behave like $\\sqrt{r}$.\nOften one uses the shifted form $\\psi(x) - \\psi(x_j)$, so that $u_h(x_i) = u_i$ at the nodes.\n\nExample: on $\\Omega = (0, 1)$ with nodes $x_i = ih$, let a crack sit at a point $c$ inside $(x_k, x_{k+1})$.\nTake $\\psi(x) = H(x - c)$, with $H$ the Heaviside step, and $J = \\set{k, k+1}$.\nThen $u_h$ can jump at $c$ by $N_k(c) a_k + N_{k+1}(c) a_{k+1}$, while the mesh does not have a node at $c$.\n\nNon-example: the standard space with only the first sum is continuous on $\\Omega$.\nIt has no enrichment, $J = \\emptyset$, so it cannot jump at $c$ unless $c$ is a mesh node and the mesh is split there.\n",
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