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. Also,
Yale University, Department of Computer Science
P.O. Box 208285, New Haven, CT 06520-8285, USA.

Gundolf Haase

Johannes Kepler University of Linz
Institute of Analysis and Computational Mathematics
Altenberger Str. 69, A-4040 Linz, Austria.

Mohamed Iskandarani

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


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.

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

AMS Classification: 68W10, 65Y05, 47N40, 76D33