13/03/2015, 14:30 — 15:30 — Room P3.10, Mathematics Building
Sergej Rjasanow, Universitaet des Saarlandes
Numerical approximation of boundary integral operators
We discuss efficient numerical methods for the boundary integral formulation of various three dimensional boundary value problems for the Laplace equation, Helmholtz equation and for the system of Lamé equations. The corresponding boundary integral equations will be discretised using Galerkin method leading to a system of linear equations with a dense matrix $A$ of some dimension $N$. A naive strategy for the solution of the corresponding linear systems would need at least $O(N^2)$ arithmetical operations and memory. Methods such as fast multipole are based on explicitly given kernel approximations by degenerate kernels. In contrast to the methods mentioned, the Adaptive Cross Approximation method (ACA) generates a hierarchical low-rank approximation from the matrix itself using only few entries and without using any explicit a priori known degenerate kernel approximation. The efficiency and convergence properties of the numerical method will be illustrated for a number of different boundary value problems and for different surfaces.
