Efficient solvers for incompressible fluid flows in geosciences

Saddle point problems involving large systems of linear equations arise in a wide variety of applications in computational science and engineering. A variety of solvers have been developed for these type of problems typically with specific applications in mind. We focus on saddle point problems as they arise from incompressible fluid flow problems in geosciences. They are characterized through a spatially variable viscosity when modeling temperature dependencies (for example, in Earth mantel convection models) or moving material interfaces (for example, in subduction zones simulation and numerical volcano models). We overview some of the iterative techniques used and discuss suitable preconditioning techniques. We discuss the implementation of the schemes using the python module Escript and compare the efficiency of these schemes when applied to convection models on a parallel computer. References K. Arrow, L. Hurwicz, and H. Uzawa. Studies in linear and nonlinear programming. Stanford University Press. Stanford, CA, 1958. M. Davies, L. Gross, and H.-B. Muhlhaus. Scripting high performance earth systems simulations on the sgi altix 3700. In Proceedings of the 7th International Conference on High Performance Computing and Grid in the Asia Pacific Region, Tokyo, Japan, pages 244--251. IEEE, 2004. doi:10.1109/HPCASIA.2004.1324041 A. de Niet and W. Wubs. Two preconditioners for saddle point problems in fluid flows. International Journal for Numerical Methods in Fluids, 54:355--377, 2007. Published online 19 December 2006 in Wiley InterScience. doi:10.1002/fld.1401 H. Elman and D. Silvester. Fast nonsymmetric iterations and preconditioning for {N}avier-{S}tokes equations. SIAM Journal on Scientific Computing., 17(1):33--46, 1996. doi:10.1137/0917004 L. Gross, L. Bourgouin, A.J. Hale, and H.-B. Mühlhaus. Interface modeling in incompressible media using level sets in escript. Physics of The Earth and Planetary Interiors, 163(1--4):23--34, 2007. doi:10.1016/j.physletb.2003.10.071 M. Kameyama, A. Kageyama, and T. Sato. Multigrid iterative algorithm using pseudo-compressibility for three-dimensional mantle convection with strongly variable viscosity. Journal Computational Physics, 206(1):162--181, 2005. doi:10.1016/j.jcp.2004.11.030 A. V. Knyazev. A preconditioned conjugate gradient method for eigenvalue problems and its implementation in a subspace. In International Ser. Numerical Mathematics, v. 96, Eigenwertaufgaben in Natur- und Ingenieurwissenschaften und ihre numerische Behandlung, Oberwolfach, 1990., pages 143--154. Birkhauser Basel, 1991. L. Moresi, F. Dufour, and H.-B. Muehlhaus. Mantle convection modeling with viscoelastic/brittle lithosphere: Numerical methodology and plate tectonic modeling. Journal on Pure and Applied Geophysics, 159(10):2335--2356, 2002. doi:10.1007/s00024-002-8738-3 D. Silvester, H. Elman, D. Kay, and A. Wathen. Efficient preconditioning of the linearized {N}avier-{S}tokes equations for incompressible flow. J. Comput. Appl. Math., 128(1-2):261--279, 2001. Numerical analysis 2000, Vol. VII, Partial differential equations. doi:10.1016/S0377-0427(00)00515-X W Weisstein, E. Einstein summation. From MathWorld---A Wolfram Web Resource. http://mathworld.wolfram.com/EinsteinSummation.htm