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
13/2014
Hierarchical matrices and the High-Frequency Fast Multipole Method for the Helmholtz Equation with Decay
Maryna Kachanovska
Abstract
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 ($\mathcal{H}$-matrices), $\mathcal{H}^2$-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 $\mathcal{H}$-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.
