Heterogeneous coupling for implicitly described domains

Many applications in physics, biology or chemistry exhibit complex geometrical shapes. Often these models feature partial differential equations (PDEs) on the complex shaped domain and its surface. At the same time the domain might be time-dependent, e.g. in cell biology the shape of a cell depends on its internal state and couples back to the cell metabolism, cf. [12]. Modern imaging techniques yield high resolution data of microscopic structures and thus allow us to exploit direct simulations. Constructing suitable meshes for complex geometries is a very involved task, thus methods to decouple the computational mesh from the geometry are of great interest. In the context of Fictitious Domain Methods a wide range of methods was developed; we want to mention explicitly the Unfitted Finite Element Method [2, 13, 4], which we build upon. These methods formulate the original problem as a problem embedded in a larger domain. Different ways of incorporating the, now internal, boundary conditions are described in the literature. Examples for applications to coupled problems can be found using XFEM [9, 10], or using fictitious domain and mortar methods [1]. Many of these methods have been developed for engineering applications and are not directly applicable to biological problems as certain processes, e.g. topology changes can not be captured. An alternative class of methods uses implicit domain descriptions as level sets [17], or phase-field models [5, review paper]. Both approaches have been applied to coupled problems (e.g. [6, 18]), but due to the diffusive representation of the coupling interface these methods can lead to numerical artifacts, including spurious fluxes. In this work we present a new approach to incorporate processes on manifolds in a heterogeneous domain-decomposition framework for implicitly described geometries. Although using a level set formulation, we avoid a diffuse coupling interface by utilizing an explicit reconstruction. It uses concepts of the Unfitted Finite Element Method and can be directly applied to image data.