Introduction to spectral methods

  title={Introduction to spectral methods},
  author={Philippe Grandcl{\'e}ment},
  journal={Eas Publications Series},
  • P. Grandclément
  • Published 6 September 2006
  • Computer Science, Mathematics
  • Eas Publications Series
This proceeding is intended to be a first introduction to spectral methods. It is written around some simple problems that are solved explicitly and in details and that aim at demonstrating the power of those methods. The mathematical foundation of the spectral approximation is first introduced, based on the Gauss quadratures. The two usual basis of Legendre and Chebyshev polynomials are then presented. The next section is devoted to one dimensional equation solvers using only one domain. Three… 

The Ritz method with Lagrange multipliers

We develop a general form of the Ritz method for trial functions that do not satisfy the essential boundary conditions. The idea is to treat the latter as variational constraints and remove them


A modified preconditioner based on the approximate inverse technique is constructed and the computational cost of each iteration in solving the preconditionsed system is O(lNxNy logNx), where Nx, Ny are the grid sizes in each direction and l is a small integer.

Spectral Procedure with Diagonalization of Operators for 2D Navier-Stokes and Heat Equations in Cylindrical Geometry

We present in this paper a spectral method for solving a problem governed by Navier-Stokes and heat equations. The Fourier-Chebyshev technique in the azimuthal direction leads to a system of

A universal solution to one-dimensional oscillatory integrals

A universal quadrature method is brought forward, which adopts Chebyshev differential matrix to solve the ordinary differential equation (ODE), which can not only obtain the indefinite integral’ function values directly, but also make the system of linear equations well-conditioned for general oscillatory integrals.

Full operator preconditioning and the accuracy of solving linear systems

This work shows that FOP can improve accuracy beyond the standard limit for both direct and iterative methods, and highlights a number of topics in numerical analysis that can be interpreted as implicitly employing FOP.

The Ritz Method for Boundary Problems with Essential Conditions as Constraints

We give an elementary derivation of an extension of the Ritz method to trial functions that do not satisfy essential boundary conditions. As in the Babuska-Brezzi approach boundary conditions are

Numerical Solution of Two Dimensional Laplace’s Equation on a Regular Domain Using Chebyshev Differentiation Matrices

This work presents an efficient procedure based on Chebychev spectral collocation method for computing the 2D Laplace’s equation on a rectangular domain. The numerical results and comparison of

Numerical studies of nonlocal parabolic partial differential equations by spectral collocation method with preconditioning

In this paper, the spectral collocation method with preconditioning is applied to solve nonlocal parabolic partial differential equations. The cubic spline interpolation is implemented for

An accurate differential quadrature procedure for the numerical solution of the moving load problem

The proposed procedure is applied herein to solve the moving load problem in beams and rectangular plates and proves that the proposed approach is highly accurate and efficient.



Numerical analysis of spectral methods

Abstract : This monograph gives a mathematical analysis of spectral methods for mixed initial-boundary value problems. Spectral methods have become increasingly popular in recent years, especially

A practical guide to pseudospectral methods

1. Introduction 2. Introduction to spectral methods via orthogonal functions 3. Introduction to PS methods via finite differences 4. Key properties of PS approximations 5. PS variations/enhancements

Spectral methods for fluid dynamics

1. Introduction.- 1.1. Historical Background.- 1.2. Some Examples of Spectral Methods.- 1.2.1. A Fourier Galerkin Method for the Wave Equation.- 1.2.2. A Chebyshev Collocation Method for the Heat

Approximations spectrales de problèmes aux limites elliptiques

Les methodes spectrales sont une technique recente d'approximation de la solution d'equations aux derivees partielles par des polynomes de haut degre. Elles ont un degre de precision infini: l'ordre

Chebyshev and Fourier Spectral Methods

A computation is a temptation that should be resisted as long as possible. "

Code for producing the figures of the above illustrative example available from Lorene CVS server (directory Lorene/Codes/Spectral

  • Code for producing the figures of the above illustrative example available from Lorene CVS server (directory Lorene/Codes/Spectral

Numerical library for spectral methods in general relativity