-Matrix approximation for elliptic solution operators in cylindric domains

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

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Submission date: 19. Mar. 2001
Pages: 23
published in: East west journal of numerical mathematics, 9 (2001) 1, p. 25-58 
DOI number (of the published article): 10.1515/JNMA.2001.25
Bibtex
MSC-Numbers: 65F50, 65F30
Keywords and phrases: operator-valued sinh function, domain decomposition, pointcaré-steklov operators
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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.