Boundary treatments for multilevel methods on unstructured meshes

In applying multilevel iterative methods on unstructured meshes, the grid hierarchy can allow general coarse grids whose boundaries may be nonmatching to the boundary of the fine grid. In this case, the standard coarse-to-fine grid transfer operators with linear interpolants are not accurate enough at Neumann boundaries so special care is needed to correctly handle different types of boundary conditions. We propose two effective ways to adapt the standard coarse-to-fine interpolations to correctly implement boundary conditions for two-dimensional polygonal domains, and we provide some numerical 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 L²-approximation and H¹-stability, which are essential in the convergence analysis of the multilevel Schwarz methods. Using these results, we give a condition number bound for two-level Schwarz methods.