Preconditioning by multilevel methods with locally modified grids

Systems of grid equations that approximate elliptic boundary value problems on locally modified grids are considered. The triangulation, which approximates the boundary with second order of accuracy, is generated from an initial uniform triangulation by shifting nodes near the boundary according to special rules. This “locally modified” grid possesses several significant features: this triangulation has a regular structure, generation of the triangulation is rather fast, this construction allows the use of multilevel preconditioning (BPX-like) methods. The proposed iterative methods for solving grid elliptic boundary value problems are based on two approaches: the fictitious space method, i.e., reduction of the original problem to that in an auxiliary (fictitious) space, and the multilevel decomposition method, i.e., construction of preconditioners by decomposing functions on hierarchical grids. The convergence rate of the corresponding iterative process with the preconditioner obtained is independent of the mesh size. The construction of the grid and the preconditioning operator for the three-dimensional problem can be done in the same manner.