MULTIPLICATIVE SCHWARZ METHODS FOR PARABOLIC PROBLEMS

The class of multiplicative Schwarz methods originated from the classi cal Schwarz alternating method. It has been shown to be one of the mos t powerful methods for solving finite element or finite difference ell iptic problems. In this paper, these methods are extended to a class o f singularly perturbed equations that are encountered when discretizin g parabolic equations by implicit methods such as the backward Euler o r Crank-Nicolson schemes. Several algorithms are discussed, including one-level, two-level, and multilevel overlapping methods. The authors also study how the convergence rates depend on the time and space disc retization parameters, as well as subspace decomposition parameters su ch as the number of subregions and the number of levels to which the f inite element space is decomposed. It is shown that in the presence of a fine enough coarse mesh space the algorithms are optimal for both s ymmetric and nonsymmetric problems, i.e., the convergence rates are in dependent of all these parameters in both two and three dimensions. If the coarse mesh space is dropped, the algorithms are still optimal bu t only if the timestep and the coarse mesh size satisfy certain relati onships.