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

Ruizhi Wang, Weibing Deng

TL;DR

This work develops a stabilized unfitted space-time finite element method for parabolic problems on moving domains, using a fully coupled space-time discretization with SUPG stabilization in time and ghost-penalty stabilization to address small-cut ill-conditioning. A rigorous a priori error analysis in a discrete energy norm is provided, supported by a space-time Poincaré–Friedrichs inequality that underpins the condition-number estimates. The method achieves optimal order convergence in the energy norm and robust conditioning (κ(A) ≲ h^{-2}), with numerical experiments in 1D and 2D confirming theory and demonstrating resilience to small cuts and boundary-layer phenomena. The approach offers mesh-generation flexibility, potential for space-time adaptivity, and strong parallelization potential for moving-domain parabolic problems.

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.

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

TL;DR

This work develops a stabilized unfitted space-time finite element method for parabolic problems on moving domains, using a fully coupled space-time discretization with SUPG stabilization in time and ghost-penalty stabilization to address small-cut ill-conditioning. A rigorous a priori error analysis in a discrete energy norm is provided, supported by a space-time Poincaré–Friedrichs inequality that underpins the condition-number estimates. The method achieves optimal order convergence in the energy norm and robust conditioning (κ(A) ≲ h^{-2}), with numerical experiments in 1D and 2D confirming theory and demonstrating resilience to small cuts and boundary-layer phenomena. The approach offers mesh-generation flexibility, potential for space-time adaptivity, and strong parallelization potential for moving-domain parabolic problems.

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.

Paper Structure

This paper contains 14 sections, 13 theorems, 83 equations, 13 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Model problem
  3. Unfitted space-time FEM
  4. A priori error analysis
  5. Some auxiliary results
  6. Stability analysis of the method
  7. Energy error estimates
  8. Condition number estimates
  9. Numerical experiments
  10. Convergence studies for 1D case
  11. Convergence studies for 2D case
  12. Small cut effect studies
  13. Boundary layer problem
  14. Conclusions

Key Result

Theorem 2.1

Suppose Assumption assump2 holds. Then for every $f \in H^{-1,0}(Q)$, $g \in H^{\frac{1}{2}}(\Sigma_s)$, $u_0 \in H^1(\Omega_0)$, there exists a unique solution $u \in W$ to problem vf1.

Figures (13)

  • Figure 1: The moving domain and the initial mesh of Example \ref{['ex1']}.
  • Figure 1: Numerical errors of Example \ref{['ex1']}
  • Figure 2: Numerical errors of Example \ref{['ex2']}
  • Figure 3: The moving domain and the initial mesh of Example \ref{['ex2']}.
  • Figure 3: Numerical errors of Example \ref{['ex3']}
  • ...and 8 more figures

Theorems & Definitions (30)

  • Remark 2.1
  • Remark 2.2
  • Theorem 2.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Lemma 4.1
  • Lemma 4.2
  • Theorem 4.3
  • proof
  • ...and 20 more