Parallel Point-oriented Multilevel Methods

Abstract

Instead of the usual nodal basis, we use a generating system for the discretization of PDEs that contains not only the basis functions of the finest level of discretization but additionally the basis functions of all coarser levels of discretization. The Galerkin-approach now results in a semidefinite system of linear equations to be solved. Standard iterative GS-methods for this system turn out to be equivalent to elaborated multigrid methods for the fine grid system.

Beside Gauss-Seidel methods for the level-wise ordered semidefinite system, we study block Gauss-Seidel methods for the point-wise ordered semidefinite system. These new algorithms show basically the same properties as conventional multi-grid methods with respect to their convergence behavior and efficiency. Additionally, they possess interesting properties with respect to parallelization. Regarding communication, the number of setup steps is only dependent on the number of processors and not on the number of levels like for parallelized multigrid methods. The amount of data to be communicated, however, increases slightly. This makes our new method perfectly suited to clusters of workstations as well as to LANs and WANs with relatively dominant communication setup.