Multilevel Preconditioning Matrices and Multigrid V-Cycle Methods

The relationship between the multilevel preconditioning matrices and a natural multigrid method is outlined. The multilevel preconditioning matrices are derived by an approximate block-factorization of the stiffness matrix computed by piecewise linear nodal basis functions. The block ordering of the nodes corresponds to a certain refining procedure, which starting with a coarse triangulation of the considered plane polygonal region generates successively refined triangulations. At each step of this refining procedure we add a new group of nodes which give the corresponding block of nodes in our multilevel ordering of the final nodal set. Under this multilevel block ordering, starting from the top level our stiffness matrix is factorized approximately, where any successive Schur complement is replaced (approximated) by the stiffness matrix of the current level. Another alternative is to view this method in the framework of the classical multigrid method of V-cycle type. For this natural multigrid method it is enough to use the corresponding two-level ordering of the stiffness matrix at each discretization level. Then the smoothing procedure is naturally derived from the stiffness matrix. The main computational task in performing one smoothing step is Solution of problems, on the current level, with a matrix (the first pivot block of the stiffness matrix in the two-level ordering) , which has a condition number bounded independently on the number of levels used. The convergence properties of these iterative methods are compared.AMS Subject Classifications: 65F10, 65N20, 65N30