On the implementation of the GMRES(m) method to elliptic equations in meteorology

Abstract

In this paper we consider the relatively new preconditioned generalized minimal residual method, restarted every m iterations (GMRES( m) ), for the solution of three-dimensional elliptic equations. Large, sparse, non-symmetric matrices are involved. The particular equation of interest is the quasi -geostrophic “omega” equation, often used in meteorology to compute vertical motion. The GMRES( m) method is tested with different preconditioners for the solution of two- and three-dimensional elliptic equations. The method requires no relaxation parameters and has no restrictions on the size of the 3D grid. GMRES can be used for more types of matrices than other methods such as SOR. Numerical results show that Jacobi preconditioned GMRES( m) performs best for 3D and high resolution problems among five different preconditioners tested, while ILIJ factorization of the partial or whole matrix A, as a preconditioner, is good for 2D and low resolution problems. The SOR preconditioners for the GMRES( m) method, with optimal relaxation parameters, are not as efficient, and the best choice for the relaxation parameter in SOR preconditioning is not the same as the best choice for the simple SOR method. An algorithm for using the preconditioned GMRES( m) method is presented.