Recent Advances in Domain Decomposition Methods for the Stokes Problem

Abstract

Domain decomposition methods for the Stokes problem are developed under a more general framework, which allows both continuous and discontinuous pressure functions and more flexibility in the construction of the coarse problem. For the case of discontinuous pressure functions, a coarse problem related to only primal velocity unknowns is shown to give scalability in both dual and primal types of domain decomposition methods. The two formulations are shown to have the same extreme eigenvalues and the ratio of the two extreme eigenvalues weakly depends on the local problem size. This property results in a good scalability in both the primal and dual formulations for the case with discontinuous pressure functions. The primal formulation can also be applied to the case with continuous pressure functions and various numerical experiments are carried out to present promising features of our approach.