Domain Decomposition Methods For Large Problems Of Elasticity
Speaker:Prof. Olof Bertil Widlund
SIAM Fellow, Professor of Mathematics and Computer Science
Courant Institute, New York University
Date & Time:18 Oct 2010 (Monday) 16:00 - 17:00
Organized by:Department of Mathematics


The domain decomposition methods considered are preconditioned conjugate gradient methods designed for the very large algebraic systems of equations which often arise in finite element practice. They are designed for massively parallel computer systems and the preconditioners are built from solvers on the substructures into which the domain of the given problem is partitioned. In addition, to obtain scalability, there must be a coarse problem, with a small number of degrees of freedom for each substructure. The design of this coarse problem is crucial for obtaining rapidly convergent iterations and poses the most interesting challenge in the analysis. Results for two families of domain decomposition methods from the overlapping Schwarz and the FETI-DP/BDDC families will be discussed with a special emphasis on almost incompressible elasticity approximated by mixed finite element and mixed spectral element methods. Some of these algorithms are now used extensively at the SANDIA, Albuquerque laboratories and might be made available as public domain software.


Throughout his career, Olof Widlund has focused on numerical algorithms for partial differential equations. His primary concern has been new algorithms and mathematical tools for their analysis. For two decades, with students and other associates, he has concentrated his efforts on domain decomposition algorithms for the large linear systems of algebraic equations that arise in many computational continuum mechanics problems, for example in fluid dynamics and elasticity. These algorithms use a preconditioned conjugate gradient approach and they are designed for parallel and distributed computers. A main challenge is to overcome the potential computational bottleneck arising because the solutions of the linear systems depend on the data everywhere in the region. A research monograph, ``Domain Decomposition Methods - Algorithms and Theory'', coauthored with Andrea Toselli, was published by the Springer Verlag in 2005. In February 2006, it received the Award for Excellence in Professional and Scholarly Publications of the Association of American Publisher, in the category Mathematics and Statistics. The book contains many results developed by the seventeen doctoral students, who have completed doctoral dissertations at the Courant Institute in this field of research since 1989.

These algorithms are increasingly being accepted by the user community and their usefulness on loosely-coupled computer systems and the very largest parallel computers has been demonstrated in a substantial number of experimental studies, some of them using the PETSc system developed at the Mathematics and Computer Science Division of the Argonne National Laboratory .

In recent years, the Courant Institute research group has focused its work on FETI-DP and BDDC algorithms for elliptic systems, including those of saddle point type and on domain decomposition methods for electro-magnetics and mortar finite element methods. Over the last year, the group has also actively developed new, hybrid domain decomposition algorithms, which combine features of iterative substructuring methods and two-level overlapping Schwarz methods.