Math 2.0

Numerical analysis Definition

Spectral element method

Builds on Finite element methodSpectral methodLagrange polynomialLegendre polynomialsGaussian quadratureGalerkin method

Definition

Let Ω⊂ℝd\Omega \subset \R^d be partitioned into elements KeK_e, each the image of the reference cube K^=[−1,1]d\hat K = [-1, 1]^d under a smooth map FeF_e. Fix a polynomial degree N≥1N \ge 1. Let ξ0=−1<ξ1<⋯<ξN=1\xi_0 = -1 < \xi_1 < \cdots < \xi_N = 1 be the Gauss–Lobatto–Legendre points. These are ±1\pm 1 and the roots of PN′P_N', where PNP_N is the Legendre polynomial of degree NN. The spectral element method is the Galerkin method with three features:

  1. the discrete space consists of continuous functions whose pullback to K^\hat K on each element is a polynomial of degree at most NN in each variable;
  2. the basis on K^\hat K is the tensor product of the Lagrange polynomials at the points ξi\xi_i;
  3. all integrals are computed with the Gauss–Lobatto–Legendre quadrature rule at the same points.

Convergence is obtained mainly by increasing NN on a fixed mesh.

Remarks

The Gauss–Lobatto–Legendre rule with N+1N + 1 points is exact for polynomials of degree at most 2N−12N - 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 NN, as for spectral methods.

Example: in one dimension with N=2N = 2, the points on [−1,1][-1, 1] are −1,0,1-1, 0, 1, the roots of P2′(ξ)=3ξP_2'(\xi) = 3\xi together with ±1\pm 1. The quadrature weights are 13,43,13\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 NN, which is the Runge phenomenon, and with exact integration the mass matrix is not diagonal. This is a finite element method of high order but not a spectral element method.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

Show the path as a graph
Finite element method, concept, no evidence levelFinite element methodconceptGalerkin method, concept, no evidence levelGalerkin methodconceptGaussian quadrature, concept, no evidence levelGaussian quadratureconceptLagrange polynomial, concept, no evidence levelLagrange polynomialconceptLegendre polynomials, concept, no evidence levelLegendre polynomialsconceptSpectral method, concept, no evidence levelSpectral methodconceptSpectral element method, definition, no evidence levelSpectral element meth…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Finite element method concept
  2. Galerkin method concept
  3. Gaussian quadrature concept
  4. Lagrange polynomial concept
  5. Legendre polynomials concept
  6. Spectral method concept
  7. Spectral element method definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Spectral_element_method
About this page
Tag8PQ405CKA3
URIm2://8PQ405CKA3
Kinddefinition
WritingW0 stub
MSC65N35, 65N30, 65M70
DefinesSpectral element method
UsesFinite element method, Spectral method, Lagrange polynomial, Legendre polynomials, Gaussian quadrature, Galerkin method

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.