On the Full Multigrid Basis Transformation in Dirichlet Domain Decomposition Preconditioners: Model Problem Analysis and Numerical Experiments
The domain decomposition (DD) technique is well-suited for constructing parallel partial differential equation solvers. By means of the Additive Schwarz Method we derive and analyse Dirichlet-type DD preconditioners. Such preconditioners contain three components which are the (modified) Schur complement preconditioner, the local Dirichlet problem preconditioner, and the so-called basis transformation. Of all components, the last one plays the most crucial role.
Amongst other methods, multigrid techniques have been popular for defining this basis transformation. However, the quality of the preconditioner deteriorates as . In this paper we investigate whether the use of the full multigrid method can remedy this shortcoming.
The technical theoretical analysis has been carried out for a model problem and the full two-grid operator. In numerical experiments the full multigrid basis transformation has been tested. The analysis shows that full multigrid behaves asymptotically as the multigrid method but displays better numerical performance.