Spectral Chebyshev Collocation for the Poisson and Biharmonic Equations
Bernard Bialecki, Andréas Karageorghis · SIAM Journal on Scientific Computing · 2010
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.