Domain Decomposition (DD) Methods
Domain decomposition (or DD) methods can generally be viewed as block versions of the Gauss–Seidel or Jacobi method that in addition may exploit overlap. If in the implementation, we use exact block inverses, the resulting algorithms are not generally of optimal complexity. For small blocks (corresponding to subdomains) to make the DD method of optimal order, we need a substantial in size coarse problem, and in order to have an overall optimal complexity of the method, we need an optimal solver for the coarse problem, which generally can be achieved by a multilevel method. For subdomain problems giving rise to large blocks, the coarse problem can be considered fixed. Then, in order to end up with an overall optimal complexity method, the subdomain problems have to be solved by an optimal method, which again can be a multilevel one. In summary, with DD-type methods to end up with an overall optimal complexity algorithm, we need in some of the components (such as subdomain or coarse-grid solutions) to employ some multilevel strategy.
This chapter also covers preconditioners based on domain embedding, auxiliary space methods, as well as preconditioners for problems with (multilevel) local refinement. In some cases, the subdomain problems allow for the use of fast (direct) elliptic solvers.We provide one such solver, as well.
KeywordsDomain Decomposition Extension Mapping Iteration Matrix Schwarz Method Auxiliary Space
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