Hierarchical matrices and the High-Frequency Fast Multipole Method for the Helmholtz Equation with Decay
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Submission date: 30. Jan. 2014 (revised version: March 2014)
MSC-Numbers: 65N38, 33C10
Keywords and phrases: hierarchical matrices, bem, fast multipole method, helmholtz equation
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The solution of boundary-value problems for the Helmholtz equation with decay is required by many physical applications, in particular viscoelastodynamics and electromagnetics. The boundary integral equation method allows to reduce the dimensionality of the problem by expressing the unknown quantity with the help of a boundary integral operator of a density given on the boundary of the domain. However, the BEM discretization of boundary integral formulations typically leads to densely populated matrices. In the last three decades a new generation of data-sparse methods for the approximation of BEM matrices was designed. Among those are panel-clustering, hierarchical matrices (ℋ-matrices), ℋ2-matrices and fast multipole methods (FMM). In this work we review main concepts of data-sparse techniques. We present a description of the high-frequency fast multipole method (HF FMM) with some technical details, both for a real and complex wavenumber. A signiﬁcant part of the report is dedicated to the error analysis of the HF FMM applied to the Helmholtz equation with a complex wavenumber. We compare the performance of the multilevel high-frequency fast multipole method and ℋ-matrices for the approximation of the single layer boundary operator for the Helmholtz equation with decay. Based on these results, a simple strategy to choose between these techniques is suggested.