Spectral Element Methods

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Spectral Element Methods

Spectral methods are essentially discretization methods for the approximate solution of partial-differential equations expressed in a weak form, based on high-order Lagrangian interpolants used in conjunction with particular quadrature rules. For spectral methods it is assumed that the solution can be expressed as a series of polynomial basis functions, which can approximate the solution well in some norm as the polynomial degree tends to infinity. These smooth basis-functions usually form an L2-complete basis. For the sake of computational efficiency this basis is typically chosen to be orthogonal in a weighted inner-product.

The spectral element method is a high-order finite element technique that combines the geometric flexibility of finite elements with the high accuracy of spectral methods. This method was pioneered in the mid 1980's by Anthony Patera at MIT and Yvon Maday at Paris-VI. It exhibits several favourable computational properties, such as the use of tensor products, naturally diagonal mass matrices, adequacy to implementations in a parallel computer system. Due to these advantages, the spectral element method is a viable alternative to currently popular methods such as finite volumes and finite elements, if accurate solutions of regular problems are sought.


Main Features of the Spectral Element Method

- Lagrange interpolants shape functions
The method is implemented by using tensor-product Lagrange interpolants within each element, where the nodes of these shape functions are placed at the zeros of Legendre polynomials (Gauss-Lobatto points) mapped from the reference domain [-1, 1] x [-1, 1] to each element. For smooth functions it can be shown that the resulting interpolants converge exponentially fast as the order of the interpolant is increased.
- Gauss-Lobatto quadrature
Efficiency is achieved by using Gauss-Lobatto quadrature for evaluating elemental integrals: the quadrature points reside at the nodal points, which enables fast tensor-product techniques to be used for iterative matrix solution methods. Gauss-Lobatto quadrature results naturally in diagonal mass matrices.



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