Domain deomposition techniques are among the methods that most people use today to solve large problems coming from finite element discretizations of boundary value problems. These methods are particularly suitable for implementation on the contemporary parallel supercomputers.

I am interested in the theory and implementation of both overlapping and non-overlapping domain decomposition methods. Together with Joseph Pasciak, I have implemented a general overlapping domain decomposition preconditioner for solving second order elliptic and parabolic problems posed on three dimensional domains with general geometry. This code runs in parallel on the Intel Paragon Supercomputer and allows for great flexibility in setting up the problem to be solved, choosing a solution algorithm and local preconditioners. The groundwater flow simulator GCT 1.3, of which I am one of the main authors, needs efficient preconditioners. The reasons are because the typical sizes of practical problems of interest are very large and the equation coefficients are rough. In addition, this code runs in parallel on the Paragon and this has to be taken into account when choosing a preconditioner. Recently, in a joint work with James Bramble and Joseph Pasciak, we developed and analyzed new non-overlapping domain decomposition preconditioners with inexact subdomain solves. Our preconditioners are specially designed for such problems. We have completed a preliminary testing of the new preconditioners. The computational results fully agree with the theoretical estimates.