S. Ferraz-Leite, D. Praetorius, M. Mayr:
"Stabile Implementierung der Randelementmethode auf stark adaptierten Netzen (Supervisor: S. Ferraz-Leite, D. Praetorius)";
Report for Bachelor thesis;
We consider the lowest-order Galerkin BEM for the weakly singular
integral equation associated with the 2D Laplacian. For the assembly
of the Galerkin matrix, certain double integrals have to be computed.
Although all entries can be computed by certain anti-derivatives
(see e.g. [Maischak 1999]), numerical experiments show that the method
becomes unstable on adaptively generated meshes due to cancellation
In this work, we therefore propose to use a certain admissibility
condition to decide whether a matrix entry is computed analytically
or by use of numerical quadrature. This condition is similar to the
one used in the context of hierarchical matrices, cf. [Hackbusch 2009].
We use an approximation result from [Börm, Löhndorf, Melenk 2005]
to prove that the approximate matrix converges exponentially to the
exact Galerkin matrix as the quadrature order is increased. Numerical
experiments underline the stability of this approach.
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