A Schwarz Preconditioner for the Cubed-Sphere

A spectral element formulation of the atmospheric two-dimensional shallow water equations on the cubed-sphere is described. The equations are written in tensor form using the contravariant and covariant velocity components. A semi-implicit time discretization results in a reduced Schur complement system for the pressure. The Laplacian operator is approximated by the L2 pseudo-Laplacian arising in the PN/PN−2 spectral element formulation of the incompressible Stokes problem. The overlapping Schwarz preconditioner of Fischer, Miller, and Tufo [An overlapping Schwarz method for spectral element simulation of three-dimensional incompressible flows, in Parallel Solution of Partial Differential Equations, IMA Vol. Math. Appl. 120, Springer, New York, 2000, pp. 159–180] based on the fast diagonalization method, is extended to generalized curvilinear coordinates. To obtain a separable operator for the linear finite element tensor-product approximation within each spectral element, extrema of the inverse metric tensor and its determinant are employed. Convergence rates and parallel CPU timings are compared against a block-Jacobi preconditioner.