Approximation of Integral Operators by Variable-Order Interpolation

Authors:
Steffen Börm;Maike Löhndorf;Jens M. Melenk
Affiliations:
Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstrasse 22–26, 04103, Leipzig, Germany;Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstrasse 22–26, 04103, Leipzig, Germany;The University of Reading, Department of Mathematics, P.O. Box 220, Whiteknights RG6 6AX, 04103, Leipzig, United Kingdom
Venue:
Numerische Mathematik
Year:
2005

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Abstract

We employ a data-sparse, recursive matrix representation, so-called **-matrices, for the efficient treatment of discretized integral operators. We obtain this format using local tensor product interpolants of the kernel function and replacing high-order approximations with piecewise lower-order ones. The scheme has optimal, i.e., linear, complexity in the memory requirement and time for the matrix-vector multiplication. We present an error analysis for integral operators of order zero. In particular, we show that the optimal convergence **(h) is retained for the classical double layer potential discretized with piecewise constant functions.