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MiS Preprint
62/2004
${\cal H}$-matrix preconditioners in convection-dominated problems
Sabine Le Borne and Lars Grasedyck
Abstract
Hierarchical matrices provide a data-sparse way to approximate fully populated matrices. In this paper we exploit ${\cal H}$-matrix techniques to approximate the $LU$-decompositions of stiffness matrices as they appear in (finite element or finite difference) discretizations of convection-dominated elliptic partial differential equations.
These sparse ${\cal H}$-matrix approximations may then be used as preconditioners in iterative methods. Whereas the approximation of the matrix inverse by an ${\cal H}$-matrix requires some modification in the underlying index clustering when applied to convection-dominant problems, the ${\cal H}$-LU-decomposition works well in the standard ${\cal H}$-matrix setting even in the convection dominant case. We will complement our theoretical analysis with some numerical examples.
