A Meshfree Partition of Unity Method for Diffusion Equations on Complex Domains

[Received on xxx] We present a numerical method for solving partial differential equations on domains with distinctive complicated geometrical properties. These will be called complex domains. Such domains occur in many real world applications, for example in geology or engineering. We are however in particular interested in applications stemming from the life sciences, especially cell biology. In this area such complex domains as retrieved from microscopy images at different scales are the norm and not the exception. Therefore, geometry is expected to directly influence the physiological function of different systems, for example signalling pathways. New numerical methods able to tackle such problems in this important area of application are urgently needed. In particular, the mesh generation problem has imposed many restric-tions in the past. The approximation approach presented here for such problems is based on a promising mesh-free Galerkin method, the partition of unity method (PUM). We introduce the main approximation features, and then focus on the construction of appropriate covers as the basis of discretisations. As a main result we present an extended version of cover construction ensuring fast convergence rates in the solution process. Parametric patches are introduced as a possible way of approximating complicated boundaries without increasing the overall problem size. Finally, the versatility, accuracy and convergence behaviour of the PUM is demonstrated in several numerical examples