Domain decomposition based $${mathcal H}$$ -LU preconditioning |
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Authors: | Lars Grasedyck Ronald Kriemann Sabine Le Borne |
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Affiliation: | (1) Max-Planck-Institute for Mathematics in the Sciences, Inselstrasse 22–26, 04103 Leipzig, Germany;(2) Tennessee Technological University, Cookeville, TN 38505, USA |
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Abstract: | Hierarchical matrices provide a data-sparse way to approximate fully populated matrices. The two basic steps in the construction of an -matrix are (a) the hierarchical construction of a matrix block partition, and (b) the blockwise approximation of matrix data by low rank matrices. In this paper, we develop a new approach to construct the necessary partition based on domain decomposition. Compared to standard geometric bisection based -matrices, this new approach yields -LU factorizations of finite element stiffness matrices with significantly improved storage and computational complexity requirements. These rigorously proven and numerically verified improvements result from an -matrix block structure which is naturally suited for parallelization and in which large subblocks of the stiffness matrix remain zero in an LU factorization. We provide numerical results in which a domain decomposition based -LU factorization is used as a preconditioner in the iterative solution of the discrete (three-dimensional) convection-diffusion equation. This work was supported in part by the US Department of Energy under Grant No. DE-FG02-04ER25649 and by the National Science Foundation under grant No. DMS-0408950. |
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Keywords: | KeywordHeading" >Mathematics Subject Classification (2000) 65F05 65F30 65F50 65N55 |
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