High order numerical methods for nonlocal operators (Jens Markus Melenk)
[P6, Melenk] focuses on nonlocal operators, in particular their numerical treatment. Such operators arise naturally in a variety of fields. One area is coupled problems, e.g., in FEM-BEM coupling, where a linear PDE posed on an unbounded domain is coupled to a more complicated one in a bounded domain. This setting fits into the topics of coupled numerical methods and block-based simulations. In this Project Part we adress the specific question of preconditioning of high order FEM-BEM coupling methods on anisotropic meshes.
More generally, nonlocal operators acting on functions defined on n-dimensional domains have attracted significant attention in recent years partially because of their ability to model a great variety of diffusion processes. Fast high order numerical methods for such operators are highly desirable not only for the quantitative treatment of processes involving such operators but also to gain insight into analytical properties. For insight into structures, we will study in this Project Part the profiles of travelling waves for reaction diffusion equations with nonlocal operators. To that end, we will employ two types of numerical approaches to realize the nonlocal operator or its inverse, which is needed for efficient time-stepping schemes when numerically tracking time-dependent reaction-diffusion equations. The first approach is based on representing the discretization of the operator as an H-matrix, i.e., a blockwise low-rank matrix. The second approach is based on viewing the operator as a Dirichlet-to-Neumann operator for a degenerate elliptic equation posed on an (n+1)-dimensional domain. This latter approach makes classical volume-based techniques such as high order FEM applicable, however, at the price of increasing the spatial dimension by 1. The ambitious long term goals of this endeavor in reaction-diffusion problems are geared toward spatially multidimensional problems. Large parts of the research are done in cooperation with Franz Achleitner of [P1, Arnold] and [P10, Schöberl]. Primary external collaborators are Christian Kuehn (TU München) and Lehel Banjai (Heriot Watt University, Edinburgh).