Conservative Discontinuous Cut Finite Element Methods: Convection-Diffusion Problems in Evolving Bulk-Interface Domains

Abstract: This work entails studying unfitted finite element discretizations for convection-diffusion equations in domains that evolve in time. In particular, these partial differential equations model the evolution of the concentration of soluble surfactants in bulk-interface domains. The work in this thesis docuses on developing numerical methods which conserve the modeled physical quantities. In this work, we propose cut finite element discretizations based on the Discontinuous Galerkin framework which are both locally and globally conservative. Local conservation is achieved on so-called macro elements, and we investigate macro element partitioning of the mesh for both stationary and time-dependent domains. Additionally, we develop globally conservative methods for time-dependent problems. We analyze the proposed methods by studying the convergence of the L2-error with respect to mesh size, condition numbers of the associated linear system matrices, and the conservation error. In numerical experiments for time-dependent problems, we show that the proposed methods have optimal convergence and that the developed macro element stabilization for time-dependent problems leads to increased accuracy while retaining stable condition numbers. Moreover, the measured conservation errors verify the global conservation of the proposed methods.