Boundary treatments for multilevel methods on unstructured meshes

In applying multilevel iterative methods on unstructured meshes, the grid h ierarchy can allow general coarse grids whose boundaries may be nonmatching to the boundary of the fine grid. In this case, the standard coarse-to-fin e grid transfer operators with linear interpolants are not accurate enough at Neumann boundaries so special care is needed to correctly handle differe nt types of boundary conditions. We propose two effective ways to adapt the standard coarse-to-fine interpolations to correctly implement boundary con ditions for two-dimensional polygonal domains, and we provide some numerica l examples of multilevel Schwarz methods (and multigrid methods) which show that these methods are as efficient as in the structured case. In addition , we prove that the proposed interpolants possess the Local optimal L-2-app roximation and H-1-stability, which are essential in the convergence analys is of the multilevel Schwarz methods. Using these results, we give a condit ion number bound for two-level Schwarz methods.