Summary
The goal of this paper is the construction of a data-sparse approximation to the Schur complement on the interface corresponding to FEM and BEM approximations of an elliptic equation by domain decomposition. Using the hierarchical (ℌ-matrix) formats we elaborate the approximate Schur complement inverse in an explicit form. The required cost \(\mathcal{O}\)(N Γ logq N Γ ) is almost linear in N Γ — the number of degrees of freedom on the interface. As input, we use the Schur complement matrices corresponding to subdomains and represented in the ℌ-matrix format. In the case of piecewise constant coefficients these matrices can be computed via the BEM representation with the cost \(\mathcal{O}\)(N Γ logq N Γ ), while in the general case the FEM discretisation leads to the complexity O(N Ω logq N Ω ).
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References
M. Costabel. Boundary integral operators on lipschitz domains: elementary results. SIAM J. Math. Anal., 19:613–625, 1988.
L. Grasedyck and W. Hackbusch. Construction and arithmetics of ℌ-matrices. Computing, 70:295–334, 2003.
W. Hackbusch. Integral equations. Theory and numerical treatment, volume 128 of ISNM. Birkhäuser, Basel, 1995.
W. Hackbusch. Direct domain decomposition using the hierarchical matrix technique. In I. Herrera, D. Keyes, O. Widlund, and R. Yates, editors, DDM14 Conference Proceedings, pages 39–50, Mexico City, Mexico, 2003. UNAM.
W. Hackbusch, B. Khoromskij, and R. Kriemann. Hierarchical matrices based on a weak admissibility criterion. Technical Report 2, MPI for Math. in the Sciences, Leipzig, 2003, submitted.
G. Hsiao, B. Khoromskij, and W. Wendland. Preconditioning for boundary element methods in domain decomposition. Engineering Analysis with Boundary Elements, 25:323–338, 2001.
B. Khoromskij and M. Melenk. Boundary concentrated finite element methods. SIAM J. Numer. Anal., 41:1–36, 2003.
B. Khoromskij and G. Wittum. Numerical solution of elliptic differential equations by reduction to the interface. Number 36 in LNCSE. Springer, 2004.
U. Langer and O. Steinbach. Boundary element tearing and interconnecting methods. Computing, To appear, 2003.
W. Wendland. Strongly elliptic boundary integral equations. In A. Iserles and M. Powell, editors, The state of the art in numerical analysis, pages 511–561, Oxford, 1987. Clarendon Press.
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Hackbusch, W., Khoromskij, B.N., Kriemann, R. (2005). Direct Schur Complement Method by Hierarchical Matrix Techniques. In: Barth, T.J., et al. Domain Decomposition Methods in Science and Engineering. Lecture Notes in Computational Science and Engineering, vol 40. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-26825-1_61
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DOI: https://doi.org/10.1007/3-540-26825-1_61
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22523-2
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