An overlapping Schwarz preconditioner for a spectral-element method on the cubed-sphere

Stephen J. Thomas

National Center for Atmospheric Research, 1850 Table Mesa Dr, Boulder CO 80305

Henry M. Tufo, Paul F. Fischer

Argonne National Laboratory, Argonne IL 60439


Spectral element formulations of the atmospheric 2-D shallow-water equations and 3-D primitive equations on the cubed-sphere are described. The equations are written in generalized curvilinear coordinates using contravariant/covariant components and the metric tensor. A semi-implicit time discretization results in a Helmholtz problem for the pressure. The Laplacian operator is approximated by the L_2 pseudo-Laplacian arising in the P_N/P_N-2 spectral element formulation of the incompressible Stoke's problem. The overlapping Schwarz preconditioner of Fischer and Tufo (1998), based on the fast diagonalization method (FDM) and scalable coarse grid solver, is extended to generalized curvilinear coordinates. To obtain a separable operator for the linear finite-element tensor-product approximation within each spectral element, the maximum value of the metric tensor and determinant are employed. Results are presented for the NCAR shallow water test suite and the Held-Suarez (1994) idealized climate simulations. Convergence rates and CPU timings are compared against a dense spectral element block-Jacobi preconditioner.