An overlapping Schwarz preconditioner for a spectral element
atmospheric model on the cubed-sphere
National Center for Atmospheric Research
Spectral element formulations of the atmospheric 2-D shallow-water equations on the
cubed-sphere are described. The equations are written in generalized curvilinear coordinates
using contravariant/covariant components following Rancic, Purser and Mesinger (1996).
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 two-level 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 minimum
of the inverse metric tensor and the maximum of its determinant are
employed. Convergence rates and parallel CPU timings on an IBM SP
are compared against a block-Jacobi preconditioner.