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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
18/2001

${\cal H}$-Matrix approximation for elliptic solution operators in cylindric domains

Ivan P. Gavrilyuk, Wolfgang Hackbusch and Boris N. Khoromskij

Abstract

We develop a data-sparse and accurate approximation of the normalised hyperbolic operator sine family generated by a strongly P-positive elliptic operator. In preceding papers, a class of H-matrices has been analysed which are data-sparse and allow an approximate matrix arithmetic with almost linear complexity. An H-matrix approximation to the operator exponent with a strongly P-positive operator was proposed by one of the authors. In the present paper, we apply the H-matrix techniques to approximate the elliptic solution operator on cylindric domains associated with an elliptic operator. It is explicitly presented by the operator-valued normalised hyperbolic sine function.

Starting with the Dunford-Cauchy representation for the hyperbolic sine operator, we then discretise the integral by the exponentially convergent quadrature rule involving a short sum of resolvents. The latter are approximated by the H-matrix techniques. Our algorithm inherits a two-level parallelism with respect to both the computation of resolvents and the treatment of different values of the spatial variable.

The approach is applied to elliptic partial differential equations in domains composed of rectangles or cylinders. In particular, we consider the H-matrix approximation to the interface Poincaré-Steklov operators with application in the Schur-complement domain decomposition method.

Received:
Mar 19, 2001
Published:
Mar 19, 2001
MSC Codes:
65F50, 65F30
Keywords:
operator-valued sinh function, domain decomposition, pointcaré-steklov operators

Related publications

inJournal
2001 Repository Open Access
Ivan P. Gavrilyuk, Wolfgang Hackbusch and Boris N. Khoromskij

\( \mathscr{H} \)-matrix approximation for elliptic solution operators in cylinder domains

In: East west journal of numerical mathematics, 9 (2001) 1, pp. 25-58