A Stabilized Unfitted Space-time Finite Element Method for Parabolic Problems on Moving Domains
This work addresses computational challenges in fluid-structure interaction or phase change simulations, but it is incremental as it builds on existing unfitted FEM approaches with specific stabilizations.
The paper tackles parabolic problems on moving domains by proposing a stabilized unfitted space-time finite element method, achieving an optimal convergence rate in error estimates and validating it with numerical examples.
This paper presents a space-time finite element method (FEM) based on an unfitted mesh for solving parabolic problems on moving domains. Unlike other unfitted space-time finite element approaches that commonly employ the discontinuous Galerkin (DG) method for time-stepping, the proposed method employs a fully coupled space-time discretization. To stabilize the time-advection term, the streamline upwind Petrov-Galerkin (SUPG) scheme is applied in the temporal direction. A ghost penalty stabilization term is further incorporated to mitigate the small cut issue, thereby ensuring the well-conditioning of the stiffness matrix. Moreover, an a priori error estimate is derived in a discrete energy norm, which achieves an optimal convergence rate with respect to the mesh size. In particular, a space-time Poincare-Friedrichs inequality is established to support the condition number analysis. Several numerical examples are provided to validate the theoretical findings.