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Nonnested multigrid methods are proved to be optimal-order solvers for finite element equations arising from elliptic problems in the presence of singularities caused by re-entrant corners and abrupt changes in the boundary conditions, where the multilevel grids are appropriately refined near singularities and are not necessarily nested. Therefore, optimal and realistic finer grids (compared with nested local refinements) could be used because of the freedom in generating nonnested multilevel grids.doi:10.1090/s0025-5718-1990-1035947-0 fatcat:2rqakqbrufcefaafwibobegb6u