Title: Rigorously Justified Robust Preconditioners for High-contrast Diffusion Equation

Speaker: Burak Aksoylu, Dept of Mathematics and the Center for Computation and Technology at LSU

Date/Time: February 4, 2008 at 10:00 - 11:00am (MST) 9:00 - 10:00 am (PT)

Location: CSRI Building/Room 90 Sandia NM Videoconferenced to Sandia CA 915/N237

Brief Abstract: We present a new preconditioner suitable for solving linear systems arising from finite element approximations of elliptic PDEs with high-contrast coefficients. The construction of the preconditioner consists of two phases. The first phase is an algebraic one which partitions the degrees of freedom into ``high'' and ``low'' permeability regions which may be of arbitrary geometry. This partition yields a corresponding blocking of the stiffness matrix and hence a formula for the action of its inverse involving the inverses of both the high permeability block and its Schur complement in the original matrix. The structure of the required sub-block inverses in the high contrast case is revealed by a singular perturbation analysis (with the contrast playing the role of a large parameter). This shows that for high enough contrast each of the sub-block inverses can be approximated well by solving only systems with constant coefficients. The second phase of the algorithm involves the approximation of these constant coefficient systems using multigrid methods. The result is a general method of algebraic character which (under suitable hypotheses) can be proved to be robust with respect to both the contrast and the mesh size.

While a similar performance is also achieved in practice by algebraic multigrid (AMG) methods, this performance is still without theoretical justification. Since the first phase of our method is comparable to the process of identifying weak and strong connections in conventional algebraic multigrid algorithms, our theory provides to some extent a theoretical justification for these successful algebraic procedures. We demonstrate the advantageous properties of our preconditioner using experiments on model problems. Our numerical experiments show that for sufficiently high contrast the performance of our new preconditioner is almost identical to that of the Ruge and St\"{u}ben AMG preconditioner, both in terms of iteration count and CPU-time.

Speaker Bio: Burak Aksoylu is an assistant professor in a joint position between the Department of Mathematics and the Center for Computation and Technology at LSU. His research concentrates on the design, analysis, and implementation of provably good numerical methods. In particular, his expertise is in multiresolution preconditioning techniques in 3D adaptive refinement settings. He has applied such multiresolution techniques to a wide array of application fields which include the simulation of electrostatics in biomolecules, digital geometry processing in the form of surface parametrization and remeshing, and multiphase flow problems in reservoir and groundwater simulations. He has ongoing efforts in constructing robust algebraic preconditioners for high-contrast media particularly for porous media flow and has experience in provably good optimal hierarchical preconditioners especially coupled with realistic 3D adaptive refinement routines.

CSRI POC: Mike Parks (1414) 845-0512