A Stabilized Unfitted Space-time Finite Element Method for Parabolic Problems on Moving Domains

TL;DR

This paper introduces a stabilized, fully coupled space-time finite element method for parabolic problems on moving domains, employing SUPG stabilization and ghost penalties to ensure accuracy and well-conditioning.

Contribution

It develops a novel stabilized unfitted space-time FEM with a priori error estimates and condition number analysis for parabolic problems on moving domains.

Findings

Achieves optimal convergence rates in a discrete energy norm.

Demonstrates stability and accuracy through numerical examples.

Provides a space-time Poincare-Friedrichs inequality for analysis.

Abstract

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.