Bialecki, Bernard; Karageorghis, Andreas Spectral Chebyshev collocation for the Poisson and biharmonic equations. (English) Zbl 1217.65213 SIAM J. Sci. Comput. 32, No. 5, 2995-3019 (2010). Summary: 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. Cited in 4 Documents MSC: 65N35 Spectral, collocation and related methods for boundary value problems involving PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35J40 Boundary value problems for higher-order elliptic equations 65F10 Iterative numerical methods for linear systems 65F08 Preconditioners for iterative methods Keywords:spectral collocation; biharmonic equations; Chebyshev polynomials; Poisson equation; Dirichlet problems; Schur complement; biconjugate gradient method; preconditioner; numerical results Software:FFTPACK PDFBibTeX XMLCite \textit{B. Bialecki} and \textit{A. Karageorghis}, SIAM J. Sci. Comput. 32, No. 5, 2995--3019 (2010; Zbl 1217.65213) Full Text: DOI