The Partition of Unity Meshfree Method for Solving Transport-Reaction Equations on Complex Domains: Implementation and Applications in the Life Sciences

Abstract

There is a wide range of highly significant scientific problems which on appropriate physical scales can be formulated as partial differential equations defined on so-called complex domains. Such complex domains often occur when material is transported through an environment of high geometrical complexity, for example porous media, domains with many obstacles, or membrane systems that are folded in a topologically complex configuration. The latter often occurs in cell biology, where the biological membranes inside the cell are strikingly topologically complex. In addition the medium in which, for example, proteins diffuse in the cell nucleus, is a complex porous media type of environment as many macro-molecules and protein-DNA complexes like the chromatin form a highly irregular structure in which many bio-molecular interactions occur. The distribution of biomolecules inside cells and tissues, their over-abundance or absence in metabolism, signalling etc., is the cause of many human diseases, therefore numerical simulations will be essential for future diagnostic abilities. Under appropriate assumptions the resulting molecular transport can be formulated as a PDE (Partial Differential Equation). The first challenge for any numerical discretisation is the generation of a cover for the underlying computational domain. Here, the mesh free Partition of Unity Method (PUM) offers a number of new degrees of freedom, as patches can be shifted, their size increased or diminished, with no need to create a non-overlapping cover at all times as is characteristic for traditional Finite Element and Finite Volume discretisations. Further advances in cover creation algorithms as discussed in this paper will allow the routine simulation of problems on domains with more complex geometries than have been treatable before.