A Domain Decomposition Preconditioner for a Parallel Finite Element Solver on Distributed Unstructured Grids

We consider a number of practical issues associated with the parallel distributed memory solution of elliptic partial diierential equations using unstructured meshes in two dimensions. The rst part of the paper describes a parallel mesh generation algorithm which is designed both for eeciency and to produce a well-partitioned, distributed mesh, suitable for the eecient parallel solution of an elliptic p.d.e. The second part of the paper concentrates on parallel domain decomposition preconditioning for the linear algebra problems which arise when solving such a p.d.e. on the unstructured meshes that we generate. It is demonstrated that by allowing the mesh generator and the p.d.e. solver to share a certain coarse grid structure we are able to obtain eecient parallel solutions to a number of large problems. Although the work is presented here in a nite element context, the issues of mesh generation and domain decomposition are not of course strictly dependent upon this particular discretization strategy.