An Additive Schwarz Preconditioner for the Spectral Element Ocean Model Formulation of the Shallow Water Equations

Craig C. Douglas

University of Kentucky
Department of Computer Science
325 McVey Hall - CCS
Lexington, KY 40506-0045, USA

Gundolf Haase

Johannes Kepler University Linz,
Institute for Analysis and Computational Mathematics
Department of Computational Mathematics and Optimization
Altenberger Strasse 69
A-4040 Linz, Austria

Mohamed Iskandarani

Rosenstiel School of Marine and Atmospheric Science 4600 Rickenbacker Causeway Miami, FL 33149-1098


Abstract

We discretize the shallow water equations with an Adams-Bashford scheme combined with the Crank-Nicholson scheme for the time derivatives and spectral elements for the discretization in space. The resulting coupled system of equations will be reduced to a Schur complement system with a special structure of the Schur complement. This system can be solved with a preconditioned conjugate gradients, where the matrix-vector product is only implicitly given. We derive an overlapping block preconditioner based on additive Schwarz methods for preconditioning the reduced system.

Key words: Shallow water equations, h-p finite elements, adaptive grids, multigrid, parallel computing, conjugate gradients, additive Schwarz preconditioner.