Multilevel preconditioners for elliptic problems by substructuring

Abstract

We consider the construction of preconditioners for reduced Schur problems for solving finite-element elliptic equations which are obtained by block Gaussian elimination of the unknowns corresponding to a number of subregions. The preconditioners are based on approximations of the considered elliptic problem on a sequence of meshes and are derived in a purely algebraic manner. It is proved that the proposed algebraic multilevel preconditioner is of nearly optimal order, that is, its relative condition number with respect to the corresponding reduced Schur complement is proportional to l 2, where l is the number of discretization levels used. This condition number is bounded uniformly with respect to the number of subregions, the way the region is partitioned (for example, into strips or into boxes), and the discontinuity of the coefficients of the elliptic operator if they are constants on the elements from the initial triangulation. The data transport, except on the coarsest level, is of local type and is proportional to the number of nodes on the interface boundary.