A leading idea is to formulate the problems at hand as Fredholm integral equations of the second kind with compact integral operators. In our opinion, this strategy is superior whenever it applies. Large-scale boundary value problems on smooth domains have been efficiently solved using such techniques ever since the introduction of the fast multipole method in the 1980s. A list of active researchers can be found here. Solvers for electromagnetic applications have reached commercialization.
In material science, however, the situation is different. Problems of interest exhibit: multiply connected domains; non-smooth, open, branching, or almost touching boundaries; mixed boundary conditions; contact inequalities; and solutions that only exist as certain limits. Suitable integral equations are rarely available in the literature. Extending the class of problems for which integral equation methods do apply lies at the heart of our work.
Our efforts span a wide range of fields: potential theory, singular integral equations, numerical quadrature, stability of discretization schemes, numerical linear algebra, fracture mechanics, homogenization, optics and imaging. Our group is small and the kind of work we do is unique in Europe. Given these circumstances, I am pleased to look back on a lustrum of exciting progress. The most important results are:
1) A higher-order scheme for quasi-static crack growth simulations with controlled accuracy, constructed in 2007: (link to paper)
2) A method for the efficient solution of integral equations on non-smooth boundaries. This method, dubbed "recursive compressed inverse preconditioning" and developed in six papers 2008-2011, has already gained recognition beyond the usual. Links: the original idea for Laplace's equation (paper 1), extension to the biharmonic equation (paper 2), further improvement and extension to mixed boundary value problems (paper 3), application to random checkerboard problems (paper4, paper5), and generalization to singular integral equations with non-zero indices (paper6).
See also:
Last modified: Thu Aug 25 11:08:19 CEST 2011