Fast Algorithms for the Discretization of Integral Equations

Description

The present project was supported in part under the TMR network "Multiscale Methods in Numerical Analysis" of the EC by the Swiss government under grant number BBW 97.0404.

Many three-dimensional problems in possibly unbounded domains can be reduced to integral equations on the boundary of the domain. The discretization of these equations by finite elements on the boundary leads to the so-called Boundary Element Method (BEM) which has, in recent years, become a widely used tool in engineering. Its competitiveness has, however, been moderate through the substantial quadrature work necessary to generate the dense stiffness matrices.

The recently developed fast algorithms, such as MULTIPOLE, PANEL CLUSTERING and also WAVELET-BASED METHODS allow to decrease the computational complexity of BEM by several orders of magnitude. While the PANEL CLUSTERING is more resistant to complex geometries (see images on the right: geometry 1, geometry 2) than the WAVELET-BASED METHODS the latter allow better preconditioning. With a new Haar wavelet basis constructed by agglomeration we combine the robustness of the PANEL CLUSTERING with the advantage of the WAVELET-BASED METHODS. The basis transformation is an O(N) algorithm and allows WAVELETS with non connected support (Wavelet1a, Wavelet1b) as in SS. With the new Haar basis at hand we implemented in the object oriented programming framework a preconditioner for weakly singular boundary integral equations of the first kind.

For different preconditioning parameters we solved the interior Dirichlet problem on various geometries. For geometry 2 the result is shown in the figure on the right. There the number of GMRes steps (z-axis) is plotted with respect to the preconditioning parameter t (y-axis) and the number of unknowns (x-axis). t equal to 0 corresponds to no preconditioning. The optimal behaviour is obtained for t equal to -0.5 which was expected by the theory.

Contacts

Prof. Christoph Schwab and Dr. Gregor Schmidlin.