On the implementation of the Chebyshev spectral method for two-dimensional elliptic equations with mixed derivatives

The issues of constructing numerical algorithms based on the Chebyshev spectral method for approximate solution of elliptic equations with mixed derivatives in a rectangular domain with homogeneous Dirichlet boundary conditions are considered. To implement the spectral method, the biconjugate gradients stabilized method with preconditioners in the form of finite difference or spectral analogs of the Laplace operator is used. A comparison of the efficiency of processing the preconditioner using the iterative method of alternating directions and the Bartels–Stewart algorithm is carried out. The presented results show that the considered algorithms demonstrate computational characteristics comparable in computation time on grids of the same dimension with the characteristics of difference methods, but they are many times superior to the latter in accuracy in the case of sufficiently smooth solutions.