Computational Issues and Parallelization

In this chapter, we discuss several computational issues that arise with the implementation of domain decomposition algorithms. The first issue concerns the choice of a decomposition of a domain into non-overlapping or overlapping subdomains. When an algorithm is implemented using multiple processors, the number of interior unknowns per subdomain must be approximately the same, to ensure load balancing, while the number of boundary unknowns must be minimized to reduce inter-subdomain communication. We describe graph partitioning algorithms which partition a grid. The second issue concerns the expected parallel computation time and speed up when implementing a domain decomposition preconditioner on an idealized parallel computer architecture. We outline heuristic estimate for this using idealized models for the computational time and inter-processor data transfer times.