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  "body": "\n## Definition\n\nLet $\\Omega \\subset \\R^d$ be partitioned into elements $K_e$, each the image of the reference cube $\\hat K = [-1, 1]^d$ under a smooth map $F_e$.\nFix a polynomial degree $N \\ge 1$.\nLet $\\xi_0 = -1 \u003c \\xi_1 \u003c \\cdots \u003c \\xi_N = 1$ be the Gauss–Lobatto–Legendre points.\nThese are $\\pm 1$ and the roots of $P_N'$, where $P_N$ is the [[Legendre polynomial|QFYGJVA996]] of degree $N$.\nThe *spectral element method* is the [[Galerkin method|Y5B0PEBKE8]] with three features:\n\n1. the discrete space consists of continuous functions whose pullback to $\\hat K$ on each element is a polynomial of degree at most $N$ in each variable;\n2. the basis on $\\hat K$ is the tensor product of the [[Lagrange polynomials|Y7KSGE97GK]] at the points $\\xi_i$;\n3. all integrals are computed with the Gauss–Lobatto–Legendre [[quadrature|FP37BTP93K]] rule at the same points.\n\nConvergence is obtained mainly by increasing $N$ on a fixed mesh.\n\n## Remarks\n\nThe Gauss–Lobatto–Legendre rule with $N + 1$ points is exact for polynomials of degree at most $2N - 1$ in one variable.\nSince the nodal basis functions vanish at all nodes but one, the quadrature makes the mass matrix diagonal.\nFor smooth solutions the error decreases faster than any power of $N$, as for [[spectral methods|MYV657YRDW]].\n\nExample: in one dimension with $N = 2$, the points on $[-1, 1]$ are $-1, 0, 1$, the roots of $P_2'(\\xi) = 3\\xi$ together with $\\pm 1$.\nThe quadrature weights are $\\tfrac13, \\tfrac43, \\tfrac13$, which is Simpson's rule.\nThe basis on each element is the three quadratic Lagrange polynomials at these points.\n\nNon-example: high-degree Lagrange polynomials at equally spaced points on each element fail clause 2.\nTheir interpolation is badly conditioned for large $N$, which is the Runge phenomenon, and with exact integration the mass matrix is not diagonal.\nThis is a [[finite element|98GZ9DVHTA]] method of high order but not a spectral element method.\n",
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