Modul: MAT870 Zurich Colloquium in Applied and Computational Mathematics

## Fast Boundary Element Methods for Electrostatic Field Computations

Prof. Dr. Steffen Börm talk

**Date:** 11.12.19 **Time:** 16.15 - 17.45 **Room:** ETH HG E 1.2

We consider the computation of electrostatic potentials by the
boundary element method. In order to obtain O(h²) convergence of
discrete solutions, we have to employ piecewise linear basis
functions and piecewise quadratic parametrizations of the surface.

Constructing the data-sparse approximations of the integral
operators required for high accuracies poses several challenges:
methods like ACA or GCA require the computation of individual
matrix entries, and since the supports of basis functions are
spread across multiple triangles, this computation is far more
computationally expensive than for simple discontinuous basis
functions. Alternative techniques like HCA allow us to
significantly reduce the computational work.

Another challenge is the parametrization of the curved triangles:
the Gramian and the normal vector are no longer constant on each
triangle, but have to be computed in each quadrature point, and
this increases the necessary work even further.

Combining efficient quadrature techniques with HCA matrix
compression, algebraic coarsening and recompression, and Krylov
solvers allows us to handle surface meshes with up to
18 million triangles on relatively affordable servers while
preserving the theoretically predicted convergence rate of the
underlying discretization scheme.