

Preprint 149/2006
Optimal Panel-Clustering in the Presence of Anisotropic Mesh Refinement
Ivan G. Graham, Lars Grasedyck, Wolfgang Hackbusch, and Stefan A. Sauter
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Submission date: 14. Dec. 2006 (revised version: December 2006)
Pages: 27
published in: SIAM journal on numerical analysis, 46 (2008) 1, p. 517-543
DOI number (of the published article): 10.1137/060677987
Bibtex
MSC-Numbers: 65N38, 65D32, 65N22
Keywords and phrases: panel-clustering, Anisotropic Meshes, boundary elements
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Abstract:
In this paper we propose and analyse a new enhanced version of the
panel-clustering algorithm for discrete boundary integral equations on
polyhedral surfaces in 3D, which is designed to perform efficiently even
when the meshes contain the highly stretched elements needed for efficient
discretisation when the solution contains edge singularities. The key
features of our algorithm are: (i) the employment of partial analytic
integration in the direction of stretching, yielding a new kernel function
on a one dimensional manifold where the influence of the high aspect ratios
in the stretched elements is removed and (ii) the introduction of a
generalised admissibility condition with respect to the partially integrated
kernel which ensures that certain stretched clusters which are inadmissible
in the classical sense now become admissible. In the context of a model
problem, we prove that our algorithm yields an accurate (up to
discretisation error) matrix-vector multiplication which requires operations, where N is the number of degrees of freedom and
is small and independent of the aspect ratio of the elements. We
also show that the classical admissibility condition leads to a sub-optimalclustering algorithm for these problems. A numerical experiment shows that
the theoretical estimates can be realised in practice. The generalised
admissibility condition can be viewed as a simple addition to the classical
method which may be useful in general when stretched meshes are present.