An iterative Chebyshev spectral solver for two-dimensional elliptic equations with variable coefficients

In this paper, we constructed and investigated two variants of iterative algorithms implemented the Chebyshev spectral method solving two-dimensional elliptic equations with variable coefficients. The considered algorithms are based on the use of the stabilised version of the bi-conjugate gradient iterative method with a combined preconditioner in the form of a diagonal matrix of equation coefficients and the discrete analogue of the Laplace operator represented by the finite difference or spectral approximations. To process the discrete analogue of the Laplace operator, we implemented the alternative direction implicit method with an optimal set of iterative parameters and the Bartels – Stewart algorithm in the first and the second cases respectively. Based on numerical experiments, we showed the high efficiency of the proposed algorithms. In both cases, the number of iterations is practically independent of the mesh size and increases moderately with an increasing degree of heterogeneity of the problem coefficients. The computational complexity of the algorithms is estimated as $O(N\sqrt{N})$, where $N$ is the number of grid points. Despite of the significant suboptimality of the computational complexity, in the case of a moderate grid size $N=n\times n, n\leq 300$, the computation time demonstrates values no worse than those of algorithms of optimal computational complexity $O(N)$.