Some Domain Decomposition Methods for Indefinite Elliptic Problems on Unstructured Meshes

In this paper, we analyze some domain decomposition methods for inde nite elliptic problems on unstructured meshes. Our theory requires no assumption on the shape of subdomains. They can have arbitrary shape. Unlike the usual domain decomposition, the coarse and ne triangulations need not be nested and quasi-uniform. Furthermore, the nodes of substructures are not necessary to be the coarse grids. This general setting in the domain decomposition makes us easier to deal with complicated geometry domain, discontinuous coe cients and boundary layers. Moreover, it becomes easier to balance the work load of each processor in a parallel computer system since we have more exibility in the choice of the number of the coarse grids. We will show that the domain decomposition methods for inde nite problems on the unstructured meshes remain optimal convergence rate when the coarse grid size and subdomain diameters are small enough.