Special Methods for PDEs
Partial differential equations (PDEs) in unbounded domains can be solved efficiently with boundary integral equations. The discretization via the boundary element collocation method (BEM) leads to large dense linear systems of equations. An implemented BEM solver for acoustic, electromagnetic, and elastic scattering problems yields superconvergence and has been applied to a variety of problems focusing on the Helmholtz equation, Maxwell's equations, and the Navier equation. Additionally, related inverse problems have been solved successfully (i.e. the Factorization method).
A recent development is mainly focusing on the efficient numerical calculation of interior transmission eigenvalues which can be used to visualize the interior of a given three dimensional object to uncover location, size, and shape of an inclusion which is the aim in non-destructive testing. However, the problem at hand is non-linear, non-elliptic, and non-selfadjoint. A novel and efficient algorithm based on the previous implemented solver has been established and is still under further development.
Another research direction is the numerical solution of PDEs involving fractional derivatives with applications to image processing. Image processing of color images (and multispectral images in general) via mathematical morphology is a recent research topic as well and software for this using the Loewner ordering for matrix fields has been developed. Additionally, a variety of adaptive filters have been constructed.