 
Summary: TOWARDS A RIGOROUSLY JUSTIFIED ALGEBRAIC PRECONDITIONER FOR
HIGHCONTRAST DIFFUSION PROBLEMS
BURAK AKSOYLU, IVAN G. GRAHAM, HECTOR KLIE, AND ROBERT SCHEICHL
Dedicated to Prof. Dr. Wolfgang Hackbusch on the occasion of his 60th birthday.
Abstract. In this paper we present a new preconditioner suitable for solving linear systems
arising from finite element approximations of elliptic PDEs with highcontrast coefficients. The
construction of the preconditioner consists of two phases. The first phase is an algebraic one
which partitions the freedoms into "high" and "low" permeability regions which may be of
arbitrary geometry. This partition yields a corresponding blocking of the stiffness matrix and
hence a formula for the action of its inverse involving the inverses of both the high permeability
block and its Schur complement in the original matrix. The structure of the required subblock
inverses in the high contrast case is revealed by a singular perturbation analysis (with the contrast
playing the role of a large parameter). This shows that for high enough contrast each of the sub
block inverses can be approximated well by solving only systems with constant coefficients. The
second phase of the algorithm involves the approximation of these constant coefficient systems
using multigrid methods. The result is a general method of algebraic character which (under
suitable hypotheses) can be proved to be robust with respect to both the contrast and the mesh
size. While a similar performance is also achieved in practice by algebraic multigrid (AMG)
methods, this performance is still without theoretical justification. Since the first phase of our
method is comparable to the process of identifying weak and strong connections in conventional
