A Cut Finite Element Method for Partial Differential Equations on Evolving Surfaces

Abstract: This thesis deals with cut finite element methods (CutFEM) for solving partial differential equations (PDEs) on evolving interfaces. Such PDEs arise for example in the study of insoluble surfactants in multiphase flow. In CutFEM, the interface is embedded in a larger mesh which need not respect the geometry of the interface. For example, the mesh of a two dimensional space containing a curve, may be used in order to solve a PDE on the curve. Consequently, in time-dependent problems, a fixed background mesh, in which the time-dependent domain is embedded, may be used.  The cut finite element method requires a representation of the interface. Previous work on CutFEM has mostly been done using linear segments to represent the interfaces. Due to the linear interface representation the proposed methods have been of, at most, second order. Higher order methods require better than linear interface representation. In this thesis, a second order CutFEM is implemented using an explicit spline representation of the interface and the convection-diffusion equation for surfactant transport along a deforming interface is solved on a curve subject to a given velocity field.  The markers, used to explicitly represent the interface, may due to the velocity field spread out alternately cluster. This may cause the interface representation to worsen. A method for keeping the interface markers evenly spread, proposed by Hou et al., is numerically investigated in the case of convection-diffusion. The method, as implemented, is shown to not be useful.