The basic approach to handle geometrically complex domains is to develop a domain decomposition version of (3)--(5). Subdomains are assumed to intersect regularly, so that grid lines are continuous across block-interfaces.
The domain decomposition algorithm considered here uses a minimal overlap and no coarse grid correction. It is known from theory [52] and experiment [13] that both constant overlap in physical space and a multi-level acceleration are required to keep the iteration count constant as the mesh is refined. Examples of constant overlap in physical space can be found in [30,44,53]. However, as observed in [26,22,13], algorithms with small overlap can be quite effective, even for large and ill-conditioned problems. Although the number of GMRES iterations is typically higher with small overlap, this is compensated for by the fact that there is less duplication of work in the overlap regions. Methods with small overlap are also much easier to implement for practical complicated problems, and tend to dominate engineering applications.
A coarse grid correction [35,12,3,4] can be quite effective for improving convergence of domain decomposition. However, in large codes used for engineering computations coarse grid correction is difficult to implement, and will not be considered here. Our present aim is to optimize efficiency of domain decomposition with minimal overlap.