Title: 

Conservative Cut Finite Element Methods

Abstract:

I will give an introduction to Cut Finite Element Methods (CutFEM) for interface problems and discuss our recent development of unfitted finite element methods that are able to conserve quantities and obtain pointwise divergence-free approximations of solenoidal velocity fields. We propose two corrections to the standard stabilization strategy in connection with CutFEM. First, we propose new stabilization terms in order to preserve the divergence-free property of Hdiv-conforming elements in case of unfitted meshes. I will present a cut finite element discretization of a Darcy interface problem with the following properties:  1) optimal rates of convergence of the approximate velocity and pressure;  2) well-posed linear systems where the condition number of the system matrix scales as it does for fitted finite element discretizations; 3) optimal rates of convergence of the approximate divergence with pointwise divergence-free approximations of solenoidal velocity fields.  All three properties hold independently of how the interface is positioned relative to the computational mesh. The second correction makes use of a macro-element partition of the mesh so that cut finite element methods based on the Discontinuous Galerkin (DG) framework can inherit properties from the DG framework. We give criteria to categorize elements in an unfitted mesh as small or large. Macro-elements can then be constructed so that each small element is connected to a large element via a chain of face neighbours. The macro-elements behave as standard finite elements and local conservation properties from the DG framework can be inherited.