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Spectral Chebyshev collocation for the Poisson and biharmonic equations. (English) Zbl 1217.65213

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.

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

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