 
Summary: ARTICLE IN PRESS APNUM:2172
Please cite this article in press as: B. Aksoylu, H. Klie, A family of physicsbased preconditioners for solving elliptic equations on highly
heterogeneous media, Applied Numerical Mathematics (2008), doi:10.1016/j.apnum.2008.06.002
JID:APNUM AID:2172 /FLA [m3SC+; v 1.95; Prn:31/07/2008; 19:05] P.1 (128)
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A family of physicsbased preconditioners for solving elliptic
equations on highly heterogeneous media
Burak Aksoylu a
, Hector Klie b,
a Department of Mathematics and Center for Computation and Technology, Louisiana State University, USA
b Center for Subsurface Modeling, Institute for Computational Science and Engineering, The University of Texas at Austin, USA
Abstract
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. However, the underlying physics is utilized to distinguish between high and low degrees of freedom which is central to
the construction of the proposed preconditioner. Namely, 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. So,
the proposed preconditioner is essentially physicsbased because without the utilization of underlying physics such an algebraic
