Approximation of Integral Operators by Variable-Order Interpolation

revised version: June 2004
Jens Markus Melenk, Steffen Börm, and Maike Löhndorf
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Submission date: 09. Sep. 2002
Pages: 40
published in: Numerische Mathematik, 99 (2005) 4, p. 605-643 
DOI number (of the published article): 10.1007/s00211-004-0564-3
MSC-Numbers: 45B05, 65N38, 68P05
Keywords and phrases: hierarchical matrices, bem, fast matrix-vector multiplication
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Abstract:

We employ a data-sparse, recursive matrix representation, so-called H²-matrices, for the efficient treatment of discretized integral operators. The format is obtained 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 mapping L² to L². In particular, we show that the optimal convergence O(h) is retained for the classical double layer potential discretized with piecewise constant functions.