Frederic Nataf : September 23, 2004

Optimized interface conditions in domain decomposition methods in the case of extreme contrasts in the coefficients.

Abstract

When the coefficients of a problem have jumps of several orders of magnitude and are anisotropic, many preconditioners and domain decomposition methods suffer from plateau in the convergence due to the presence of very small isolated eigenvalues in the spectrum of the preconditioned linear system. One way to improve the preconditioner is to use a linear algebra technique called deflation, or very similarly to use coarse grid corrections. In both cases, it is necessary to identify and compute, at least approximately, all the eigenvectors corresponding to the "bad" eigenvalues.

In the framework of domain decomposition methods, we propose a way to design interface conditions so that convergence is fast and does not plateau. The method relies only on the knowledge of the smallest and largest eigenvalues of an auxiliary matrix. The eigenvectors are not used. The method relies on Van der Sluis' result on a quasi-optimal diagonal preconditioner for a symmetric positive definite matrix. It is then possible to design Robin interface conditions using only one real parameter for the entire interface. By adding a second real parameter and more general interface conditions, it is possible to take into account highly heterogeneous media. A first analysis is made at the semi-discrete level (i.e. the equation is kept continuous in the direction normal to the interface). A second analysis is made at the ``fully'' discrete level. Numerical results are given for both analyses and are compared with other approaches.