We perform a new analysis of the preconditioned matrix, which shows rather explicitly how its condition number depends on the variable coefficient in the PDE as well as on the coarse mesh and overlap parameters. The classical estimates for this preconditioner with linear coarsening guarantee good conditioning only when the coefficient varies mildly inside the coarse grid elements. By contrast, our new results show that, with a good choice of subdomains and coarse space basis functions, the preconditioner can still be robust even for large coefficient variation inside domains, when the classical method fails to be robust. In particular our estimates prove very precisely the previously made empirical observation that the use of low-energy coarse spaces can lead to robust preconditioners. We go on to consider coarse spaces constructed (a) via smoothed aggregation and (b) from multiscale finite elements, and we prove that preconditioners using these types of coarsenings lead to robust preconditioners for a variety of binary (i.e. two-scale) media model problems. Moreover numerical experiments show that the new preconditioners have greatly improved performance over standard preconditioners/solvers even in the random coefficient case.

The details of this work can be found in:

*R. Scheichl and E. Vainikko,***Robust Aggregation-Based Coarsening for Additive Schwarz in the Case of Highly Variable Coefficients**, in Proceedings of the European Conference on Computational Fluid Dynamics, ECCOMAS CFD 2006 (P. Wesseling, E. Onate, J. Periaux, Eds.), TU Delft, 2006. [Preprint]

*I.G. Graham and R. Scheichl,***Robust Domain Decomposition Algorithms for Multiscale PDEs**, Numerical Methods for Partial Differential Equations**23**(4):859-878, 2007. [Preprint]

*I.G. Graham, P. Lechner and R. Scheichl,***Domain Decomposition for Multiscale PDEs**, Numerische Mathematik**106**:589-626, 2007. [Preprint]

*R. Scheichl and E. Vainikko,***Additive Schwarz with Aggregation-Based Coarsening for Elliptic Problems with Highly Variable Coefficients**, Computing**80**(4):319-343, 2007. [Preprint]

*I.G. Graham and R. Scheichl,***Coefficient-explicit Condition Number Bounds for Overlapping Additive Schwarz**, in Proceedings of the 17th International Conference on Domain Decomposition Methods, Strobl, Austria, 3-7 July 2006, Langer, U., Discacciati, M. Keyes, D. et al. (Eds.), Domain Decomposition Methods in Science and Engineering XVII, Lecture Notes in Computational Science and Engineering, Vol. 60, Springer, New York, 2008. [Preprint]

*J. Van lent, R. Scheichl and I.G. Graham,***Energy Minimizing Coarse Spaces for Two-level Schwarz Methods for Multiscale PDEs**, BICS Preprint 12/08, Bath Institute for Complex Systems, University of Bath, 2008. [Preprint]

*C. Pechstein and R. Scheichl,***Analysis of FETI Methods for Multiscale PDEs**, RICAM Report 2008-20, Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, to appear in Numerische Mathematik, 2008. [Preprint]

*C. Pechstein and R. Scheichl,***Robust FETI Solvers for Multiscale Elliptic PDEs**, submitted to the Proceedings of the 6th Int. Conf. on Scientific Computing in Electrical Engineering, "Mathematics in Industry -- Scientific Computing in Electrical Engineering (SCEE 2008)", Springer, 2008. [Preprint]

*C. Pechstein and R. Scheichl,***Analysis of FETI Methods for Multiscale PDEs - Part II: Interface Variation**, in preparation.

Recently I have also started a collaboration with
*Frederic Nataf (Paris VI)*
on alternative (Robin or order-2) boundary conditions for two-level Schwarz
methods in the context of multiscale PDEs. Results to follow.

[Home] [Department of Mathematical Sciences] [University of Bath]

Last updated 26/06/2006