Abstract
Hierarchical matrices provide a data-sparse way to approximate fully populated matrices. The two basic steps in the construction of an \({{\mathcal H}}\) -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 \({{\mathcal H}}\) -matrices, this new approach yields \({\mathcal H}\) -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 \({\mathcal H}\) -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 \({{\mathcal H}}\) -LU factorization is used as a preconditioner in the iterative solution of the discrete (three-dimensional) convection-diffusion equation.
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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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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Grasedyck, L., Kriemann, R. & Le Borne, S. Domain decomposition based \({\mathcal H}\) -LU preconditioning. Numer. Math. 112, 565–600 (2009). https://doi.org/10.1007/s00211-009-0218-6
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DOI: https://doi.org/10.1007/s00211-009-0218-6