Spectral Chebyshev Collocation for the Poisson and Biharmonic Equations

Authors:
Bernard Bialecki;Andreas Karageorghis
Affiliations:
bbialeck@mines.edu;andreask@ucy.ac.cy
Venue:
SIAM Journal on Scientific Computing
Year:
2010

Citing 16
Cited 0

Efficient direct methods for solving the spectral collocation equations for stokes flow in rectangularly decomposable domains

SIAM Journal on Scientific and Statistical Computing
Finite-element preconditioning for pseudospectral solutions of elliptic problems

SIAM Journal on Scientific and Statistical Computing
BI-CGSTAB: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems

SIAM Journal on Scientific and Statistical Computing
An accurate solution of the Poisson equation by the Chebyshev collocation method

Journal of Computational Physics
Efficient spectral-Galerkin method II.: direct solvers of second- and fourth-order equations using Chebyshev polynomials

SIAM Journal on Scientific Computing
Mass- and momentum-conserving spectral methods for Stokes flow

Journal of Computational and Applied Mathematics
Matrix computations (3rd ed.)

Matrix computations (3rd ed.)
Timely Communication: Efficient Algorithms for Solving a Fourth-Order Equation with the Spectral-Galerkin Method

SIAM Journal on Scientific Computing
An accurate solution of the Poisson equation by the Chebyshev-tau method

Journal of Computational and Applied Mathematics
Reduction of a band-symmetric generalized eigenvalue problem

Communications of the ACM
A Legendre Spectral Galerkin Method for the Biharmonic Dirichlet Problem

SIAM Journal on Scientific Computing
The Ehrlich--Aberth Method for the Nonsymmetric Tridiagonal Eigenvalue Problem

SIAM Journal on Matrix Analysis and Applications
A spectral collocation method based on integrated Chebyshev polynomials for two-dimensional biharmonic boundary-value problems

Journal of Computational and Applied Mathematics
Spectral Chebyshev-Fourier collocation for the Helmholtz and variable coefficient equations in a disk

Journal of Computational Physics
A Fast Direct Solver for the Biharmonic Problem in a Rectangular Grid

SIAM Journal on Scientific Computing
Numerical Approximation of Partial Differential Equations

Numerical Approximation of Partial Differential Equations

Quantified Score

Hi-index	0.00

Visualization

Abstract

This paper is concerned with the spectral Chebyshev collocation solution of the Dirichlet problems for the Poisson and biharmonic equations in a square. The collocation schemes are solved at a cost of $2N^3+O(N^2\log N)$ operations using an appropriate set of basis functions, a matrix diagonalization algorithm, and fast Fourier transforms. For the biharmonic problem, the resulting Schur complement system is solved by a preconditioned biconjugate gradient method. An application of the Poisson spectral preconditioner is discussed for the solution of a variable coefficient spectral problem. Numerical results confirm the efficiency of the proposed algorithms and the spectral and polynomial accuracy of the collocation schemes for smooth and singular solutions, respectively.