Multiplicative Schwarz Methods for Parabolic Problems

The class of multiplicative Schwarz methods originated from the classical Schwarz alternating method. It has been shown to be one of the most powerful methods for solving finite element or finite difference elliptic problems. In this paper, these methods are extended to a class of singularly perturbed equations that are encountered when discretizing parabolic equations by implicit methods such as the backward Euler or 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 discretization parameters, as well as subspace decomposition parameters such as the number of subregions and the number of levels to which the finite element space is decomposed. It is shown that in the presence of a fine enough coarse mesh space the algorithms are optimal for both symmetric and nonsymmetric problems, i.e., the convergence rates are independent of all these parameters in both two and three dimensions. If the coarse mesh space is dropped, the algorithms are still optimal but only if the timestep and the coarse mesh size satisfy certain relationships.