Spectral Methods

MOX 16
Spectral Methods
Wednesday 5th March 2003
Canuto, Claudio; Quarteroni, Alfio
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Spectral methods represent a family of methods for the numerical approximation of partial differential equations. Their common denominator is to rely on high order polynomial expansions, notably trigonometric polynomials for periodic problems, orthogonal Jacobi polynomials for non periodic boundary value problems. They have the potential of providing high rate of convergence when applied to problems with regular data. They can be regarded as members of the broad family of (generalized) Galerkin methods with munerical evaluation of integrals based on Gaussian nodes. In a first part we introduce the methods on a computational domain of simple shape, analyze their approximation properties as well as their algorithmic features. Next we address the issue of how these methods can be extended to more complex geometrical domains be retaining their distinctive approximation properties.