---
tag: 8PQ405CKA3
kind: definition
name: Spectral element method
msc: [65N35, 65N30, 65M70]
defines: T1Y1CJR1JB
uses: [98GZ9DVHTA, MYV657YRDW, Y7KSGE97GK, QFYGJVA996, FP37BTP93K, Y5B0PEBKE8]
sources:
  - url: https://en.wikipedia.org/wiki/Spectral_element_method
---

## Definition

Let $\Omega \subset \R^d$ be partitioned into elements $K_e$, each the image of the reference cube $\hat K = [-1, 1]^d$ under a smooth map $F_e$.
Fix a polynomial degree $N \ge 1$.
Let $\xi_0 = -1 < \xi_1 < \cdots < \xi_N = 1$ be the Gauss–Lobatto–Legendre points.
These are $\pm 1$ and the roots of $P_N'$, where $P_N$ is the [[Legendre polynomial|QFYGJVA996]] of degree $N$.
The *spectral element method* is the [[Galerkin method|Y5B0PEBKE8]] with three features:

1. 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;
2. the basis on $\hat K$ is the tensor product of the [[Lagrange polynomials|Y7KSGE97GK]] at the points $\xi_i$;
3. all integrals are computed with the Gauss–Lobatto–Legendre [[quadrature|FP37BTP93K]] rule at the same points.

Convergence is obtained mainly by increasing $N$ on a fixed mesh.

## Remarks

The Gauss–Lobatto–Legendre rule with $N + 1$ points is exact for polynomials of degree at most $2N - 1$ in one variable.
Since the nodal basis functions vanish at all nodes but one, the quadrature makes the mass matrix diagonal.
For smooth solutions the error decreases faster than any power of $N$, as for [[spectral methods|MYV657YRDW]].

Example: 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 $\pm 1$.
The quadrature weights are $\tfrac13, \tfrac43, \tfrac13$, which is Simpson's rule.
The basis on each element is the three quadratic Lagrange polynomials at these points.

Non-example: high-degree Lagrange polynomials at equally spaced points on each element fail clause 2.
Their interpolation is badly conditioned for large $N$, which is the Runge phenomenon, and with exact integration the mass matrix is not diagonal.
This is a [[finite element|98GZ9DVHTA]] method of high order but not a spectral element method.