MULTILEVEL ADAPTIVE METHODS FOR ELLIPTIC EIGENPROBLEMS - A 2-LEVEL CONVERGENCE THEORY

The fast adaptive composite grid method (FAC) is a multilevel scheme that attempts to provide accurate and optimal computation of solutions of partial differential equations on refined grids. The central feature of FAC that ensures its efficiency is its domain-decomposition-like use of uniform grids that cover different regions with different meshsizes, but that fully overlap. This full overlap is geometric in the sense that the regions covered by the finer grids are also covered by the coarser ones, but only at the coarser grid scales. FAC's full overlap yields fast convergence rates, while its limited scales in the overlap region assure optimal complexity. This paper develops a two-level convergence theory for FAC applied to elliptic eigenproblems. The FAC method is realized by constructing a refined grid version of RQMG, which is a multilevel Rayleigh quotient minimization scheme developed in an earlier paper. The theory shows that this form of FAC converges independent of the meshsizes of either the global or local grids. As such, this is the first theory established for multilevel adaptive methods that applied to other than linear equations.