Fast Boundary Element Methods for Electrostatic Field Computations

Vortrag von Prof. Dr. Steffen Börm

Datum: 11.12.19  Zeit: 16.15 - 17.45  Raum: 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.