-matrix preconditioners in convection-dominated problems
Sabine Le Borne and Lars Grasedyck
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Submission date: 28. Sep. 2004
published in: SIAM journal on matrix analysis and applications, 27 (2006) 4, p. 1172-1183
DOI number (of the published article): 10.1137/040615845
MSC-Numbers: 65F05, 65F30, 65F50
Keywords and phrases: hierarchical matrices, preconditioning, convection-dominant problems
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Hierarchical matrices provide a data-sparse way to approximate fully populated matrices. In this paper we exploit -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 -matrix approximations may then be used as preconditioners in iterative methods. Whereas the approximation of the matrix inverse by an -matrix requires some modification in the underlying index clustering when applied to convection-dominant problems, the -LU-decomposition works well in the standard -matrix setting even in the convection dominant case. We will complement our theoretical analysis with some numerical examples.