**Andre Massing
UMIT Research Lab
Umeå University
Umeå
**

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).