A space-time cut finite element method for a time-dependent parabolic model problem

Abstract

In this thesis, a space-time finite element method for the heat equation on overlapping meshes is presented. Here, overlapping meshes means that we have a stationary mesh of the solution domain with an additional mesh that is allowed to move around in and through the solution domain. The thesis contains a derivation, an analysis, and results from an implementation of the method. The derivation starts with a strong formulation of the problem and ends with a finite element variational formulation together with adequate function spaces. For the finite element solution, we use continuous Galerkin in space and discontinuous Galerkin in time, with the addition of a discontinuity in the solution on the space-time boundary between the two meshes. In the analysis, we propose an a priori error estimate for the method with discontinuous Galerkin of order zero and one, i.e., dG(0) and dG(1). For dG(1), the error estimate indicates that the movement of the additional mesh decreases the order of convergence of the error, with respect to the time step, from the third to the second order, when the speed of the moving mesh is large enough. The order of convergence with respect to the step size for dG(1), as well as the error convergence for dG(0), are unaffected by the moving mesh and are thus as in the case with only a stationary mesh, presented in [2, 3]. An implementation of the method in one spatial dimension, with piecewise linear elements in space, and dG(0) and dG(1) in time, has also been performed. The numerical results of the implementation show the superiority of using dG(1) instead of dG(0) for overlapping meshes. The numerical results also confirm the behaviour of the error convergence, indicated by the a priori error estimate. Keywords: partial differential equation, finite element method, space-time cut, time-dependent, parabolic problem, heat equation, overlapping mesh, moving mesh, discontinuous Galerkin, a priori.