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MiS Preprint

Data-sparse approximate inverse in elliptic problems: Green's function approach

Boris N. Khoromskij


In preceding papers [15, 16, 17, 21], a class of matrices (H-matrices) has been developed which are data-sparse and allow to approximate nonlocal operators with almost linear complexity. In the present paper, a method is described for a semi-explicit H-matrix approximation to the inverse of an elliptic differential operator in Rd with piecewise smooth coefficients. The approach is based on the additive splitting to the corresponding Green's function, which is treated by H-matrices combined with the hp-FEM approximation on boundary concentrated meshes. In the case of jumping coefficients, the desired inverse operator is obtained as a direct sum of local inverses over subdomains and the global Schur-complement on the interface. As a by-product, our construction provides a data-sparse approximate inverse preconditioner for elliptic equations with variable coefficients.

[15] W. Hackbusch: A Sparse Matrix Arithmetic based on H-Matrices. Part I: Introduction to H-Matrices. Computing 62 (1999), 89-108.
[16] W. Hackbusch and B.N. Khoromskij: A sparse H-matrix arithmetic. Part II: Application to multi-dimensional problems. Computing 64 (2000), 21-47.
[17] W. Hackbusch and B. N. Khoromskij: A sparse H-matrix arithmetic: General complexity estimates. J. of Comp. and Appl. Math., 125 (2000) 479-501.
[21] W. Hackbusch, B. N. Khoromskij and S. Sauter: On H2-matrices. In: Lectures on Applied Mathematics (H.-J. Bungartz, R. Hoppe, C. Zenger, eds.), Springer Verlag, 2000, 9-29.

MSC Codes:
65F50, 65F30, 65N35, 65F10
elliptic equations, data-sparse approximate inverse, h-matrices, boundary concentrated hp-fem

Related publications

2003 Repository Open Access
Boris N. Khoromskij

Data-sparse elliptic operator inverse based on explicit approximation to the Green function

In: Journal of numerical mathematics, 11 (2003) 2, pp. 135-162