ITERATIVE METHODS FOR THE SOLUTION OF ELLIPTIC PROBLEMS ON REGIONS PARTITIONED INTO SUBSTRUCTURES.

Finite element problems can often naturally be divided into subproblems which correspond to subregions into which the region has been partitioned or from which it was originally assembled. A class of iterative methods is discussed in which these subproblems are solved by direct methods, while the interaction across the curves or surfaces which divide the region is handled by a conjugate gradient method. A mathematical framework for this work is provided by regularity theory for elliptic finite element problems and by block Gaussian elimination. A full development of the theory, which shows that certain of these methods are optimal, is given for Lagrangian finite element approximations of second order linear elliptic problems in the plane. Results from numerical experiments are also reported.