LOCAL REFINEMENT TECHNIQUES FOR ELLIPTIC PROBLEMS ON CELL-CENTERED GRIDS .3. ALGEBRAIC MULTILEVEL BEPS PRECONDITIONERS
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abstract
Algebraic multilevel analogues of the BEPS preconditioner designed for solving discrete elliptic problems on grids with local refinement are formulated, and bounds on their relative condition numbers, with respect to the composite-grid matrix, are derived. The V-cycle and, more generally, v-fold V-cycle multilevel BEPS preconditioners are presented and studied. It is proved that for 2-D problems the V-cycle multilevel BEPS is almost optimal, whereas the v-fold V-cycle algebraic multilevel BEPS is optimal under a mild restriction on the composite cell-centered grid. For the v-fold multilevel BEPS, the variational relation between the finite difference matrix and the corresponding matrix on the next-coarser level is not necessarily required. Since they are purely algebraically derived, the v-fold (v>1) multilevel BEPS preconditioners perform without any restrictionson the shape of subregions, unless the refinement is too fast. For the V-cycle BEPS preconditioner (2-D problem), a variational relation between the matrices on two consecutive grids is required, but there is no restriction on the method of refinement on the shape, or on the size of the subdomains. 1991 Springer-Verlag.
