# Helsing 200805

### From NA-Wiki

The last decades have seen huge progress on solving elliptic PDEs numerically using Fredholm second kind integral equation methods. Problems with smooth and well-separated boundaries and simple boundary conditions for Laplace's and similar equations in two dimensions are well understood in the following sense: to obtain accurate representations of the solution in terms of layer densities is often a standard task. When it comes to computing the actual solution at a large number of arbitrary points in the computational domain, however, there are still some open questions. The first part of the talk will present new results on the construction of such "field plots".

In applications, particularly in materials science, geometries of interest, such as aggregates of grains and fractured specimens, almost never have smooth or well-separated boundaries. As a consequence, second kind integral equations which otherwise are of Fredholm type lose important properties. Smooth kernels develop fixed (near) singularities. Unknown layer densities, which are to be solved for, exhibit complicated asymptotic behavior. This leads to serious degradation of the performance of standard numerical schemes. Techniques such as "special basis functions" and "mesh grading" have been suggested to overcome these difficulties. The second part of the talk presents a new technique called "recursive compressed inverse preconditioning" which we believe is superior (whenever it applies).

Reference to part one: "On the evaluation of layer potentials close to their sources" J. Comput. Phys., vol. 227(5), pp. 2899-2921, (2008) http://dx.doi.org/10.1016/j.jcp.2007.11.024