CutFEM: Discretizing Geometry and Partial Differential Equation

Andre Massing
UMIT Research Lab
Umeå University


We consider the cut finite element framework for the numerical solution of partial differential equations (PDEs) posed on complicated domains. Departed from the classical applications such as fluid flows in complex domain or fluid-structure interaction with large deformation, we will focus on PDE problems which are posed and coupled through domains of *different* topological dimensionality.

A prominent use case are flow and transport problems in porous media when large-scale networks of fractures and channels are modelled as 2D or 1D geometries embedded into a 3D bulk domain. Another important example is the modeling of cell motility where reaction-diffusion systems on the cell membrane and inner cell are coupled to describe the active reorganization of the cytoskeleton. But with complex lower-dimensional and possibly evolving geometries, traditional PDE discretization technologies are severely limited by their strong requirements on the domain discretization.

In this talk, we focus on the cut finite element framework as one possible and general approach to discretize coupled PDE systems on complex domains. To allow for a flexible discretization and easy coupling between PDEs in the bulk and on lower dimensional manifold-type domains, the lower-dimensional geometries are embedded in an unfitted manner into a three dimensional background mesh consisting of tetrahedra. Since the embedded geometry is not aligned with the background mesh, we use the trace of finite element functions defined on the tetrahedra as trial and test functions in the discrete variational formulations. As the resulting linear system may be severely ill-conditioned due to possibly small intersections between the embedded manifold and the background mesh, we discuss several possibilities for adding (weakly) consistent stabilizations terms to the original bilinear form. The proposed discretization schemes have optimal convergence properties and give rise to discrete linear systems which are well-conditioned independent of the intersection configuration. Along with presentation of the framework we will give a number of numerical examples which illustrate the theoretical findings and the applicability of the framework to complex modeling problems.

Thought as an overview over recent CutFEM technologies, the content of this talk builds on several recent and earlier collaboration projects with Erik Burman (London), Susanne Claus (Cardiff), Peter Hansbo (Jönköping), Mats G. Larson (Umeå), Anders Logg (Göteborg), Marie Rognes (Oslo), Benedikt Schott and Wolfgang Wall (Munich), Sara Zahedi (Stockholm).