 
Summary: Proceedings in Applied Mathematics and Mechanics, 28 October 2007
Physicsbased preconditioners for solving PDEs on highly heterogeneous
media
Burak Aksoylu1
and Hector Klie 2
1
Louisiana State University, Department of Mathematics and Center for Computation and Technology
2
The University of Texas at Austin, The Institute of Computational and Engineering Sciences
Eigenvalues of smallest magnitude become a major bottleneck for iterative solvers especially when the underlying physical
properties have severe contrasts. These contrasts are commonly found in many applications such as composite materials,
geological rock properties, and thermal and electrical conductivity. The main objective of this work is to construct a method
as algebraic as possible that could efficiently exploit the connectivity of highly heterogeneous media in the solution of diffu
sion operators. We propose an algebraic way of separating binarylike systems according to a given threshold into high and
lowconductivity regimes of coefficient size O(m) and O(1), respectively where m 1. The condition number of the linear
system depends both on the mesh size and the coefficient size m. For our purposes, we address only the m dependence since
the condition number of the linear system is mainly governed by the highconductivity subblock. Thus, the proposed strategy
is inspired by capturing the relevant physics governing the problem. Based on the algebraic construction, a twostage precon
ditioning strategy is developed as follows: (1) a first stage that comprises approximation to the components of the solution
associated to small eigenvalues and, (2) a second stage that deals with the remaining solution components with a deflation
