Domain Decomposition Methods Based on Cholesky's Algorithm
Olof B. Widlund (Courant Institute New York)
The often very large linear systems of algebraic equations
which arise in finite element analysis of linear elasticity
and other applications are traditionally solved directly
using a Cholesky factorization in engineering software systems.
Considerable and steady progress is being made in the deployment of
efficient solvers of this kind. In this talk, an alternative approach
will be explored, which has been proven quite successful even on
massively parallel computers.
Domain decomposition methods are preconditioned iterative methods
often using conjugate gradients. The preconditioners are often
built from direct Cholesky solvers for problems on the subdomains
and a global component which is necessary to ensure scalability,
i.e., a convergence rate which is independent of the number of
subdomains into which the original elastic body, etc., has been
divided. We will demonstrate that FETI-DP and BDDC algorithms
can be built from a few simple components of which a Cholesky
solver is the most important. This framework also highlights
the close relationship between these two families of algorithms.
We will also touch on the design of multi-level algorithms and
the extension to certain saddle point problems.
The research reported is the result of two projects conducted jointly
with Axel Klawonn and Oliver Rheinbach of the University of Essen,
Germany and with Jing Li of Kent State University.