In this paper we propose and analyse a new hierarchical Cholesky (H-Cholesky) factorization based preconditioner for iterative solving the elliptic equations with highly jumping coefficients arising in the so-called skin-modeling problem in 3D [8]. First, we construct the block-diagonal approximation to the FE stiffness matrix, which is well suited to the “perforated” structure of the coefficients. We apply the H-Cholesky factorization of this block-diagonal matrix as a preconditioner in the PCG iteration. It is shown that the new preconditioner is robust with respect to jumps in the coefficients and it requires less storage and computing time than the standard H-Cholesky factorization.
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References
M. Bebendorf. Hierarchical LU decomposition-based preconditioners for BEM. Computing, 74(3):225–247, 2005.
L. Grasedyck and S. Börm. \({\cal H}\)-matrix library: www.hlib.org.
W. Hackbusch. A sparse matrix arithmetic based on \({\cal H}\)-matrices. I. Introduction to \({\cal H}\)-matrices. Computing, 62(2):89–108, 1999.
W. Hackbusch. Direct domain decomposition using the hierarchical matrix technique. In Domain Decomposition Methods in Science and Engineering, pages 39–50. Natl. Auton. Univ. Mex., México, 2003.
W. Hackbusch and B.N. Khoromskij. A sparse \({\cal H}\)-matrix arithmetic. II. Application to multi-dimensional problems. Computing, 64(1):21–47, 2000.
W. Hackbusch, B.N. Khoromskij, and R. Kriemann. Direct Schur complement method by domain decomposition based on \({\cal H}\)-matrix approximation. Comput. Vis. Sci., 8(3-4):179–188, 2005.
B.N Khoromskij and A. Litvinenko. Domain decomposition based \({\cal H}\)-matrix preconditioner for the 2D and 3D skin problem. Technical Report 95, Max-Planck Institute for Math. in the Sciences, 2006.
B.N. Khoromskij and G. Wittum. Numerical Solution of Elliptic Differential Equations by Reduction to the Interface, volume 36 of Lecture Notes in Computational Science and Engineering. Springer-Verlag, Berlin, 2004.
S. Le Borne and L. Grasedyck. \({\cal H}\)-matrix preconditioners in convection-dominated problems. SIAM J. Matrix Anal. Appl., 27(4):1172–1183, 2006.
M. Lintner. The eigenvalue problem for the 2D Laplacian in \({\cal H}\)-matrix arithmetic and application to the heat and wave equation. Computing, 72(3-4):293–323, 2004.
A. Litvinenko. Application of Hierarchical Matrices for Solving Multiscale Problems. PhD thesis, Leipzig University, 2006.
A. Quarteroni and A. Valli. Domain Decomposition Methods for Partial Differential Equations. Oxford University Press, Oxford, 1999.
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Khoromskij, B.N., Litvinenko, A. (2008). Domain Decomposition Based H-Matrix Preconditioners for the Skin Problem. In: Langer, U., Discacciati, M., Keyes, D.E., Widlund, O.B., Zulehner, W. (eds) Domain Decomposition Methods in Science and Engineering XVII. Lecture Notes in Computational Science and Engineering, vol 60. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75199-1_17
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DOI: https://doi.org/10.1007/978-3-540-75199-1_17
Publisher Name: Springer, Berlin, Heidelberg
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