Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Discretization Error Analysis of a High Order Unfitted Space-Time Method for moving domain problems

Fabian Heimann, Christoph Lehrenfeld, Janosch Preuß

TL;DR

<3-5 sentence high-level summary>This paper analyzes a higher-order unfitted space-time DG method for convection-diffusion problems on moving domains, where the geometry is represented by isoparametric space-time mappings. It extends previous geometric error bounds to the fully discrete setting, including ghost-penalty stabilization and a mass-conserving variant, and establishes stability and coercivity through an inf-sup framework. A Strang-type decomposition yields an a priori error bound that separates geometric, interpolation, and consistency contributions, showing optimal convergence rates under mild geometric regularity and small mesh/time-step parameters. The work is significant for solving moving-domain PDEs without remeshing, enabling high-order accuracy on evolving geometries with robust mass conservation and rigorous error control.

Abstract

We present a numerical analysis of a higher order unfitted space-time Finite Element method applied to a convection-diffusion model problem posed on a moving bulk domain. The method uses isoparametric space-time mappings for the geometry approximation of level set domains and has been presented and investigated computationally in [Heimann, Lehrenfeld, Preuß, SIAM J. Sci. Comp. 45(2), 2023, B139 - B165]. Recently, in [Heimann, Lehrenfeld, IMA J. Numer. Anal., 2025] error bounds for the geometry approximation have been proven. In this paper we prove stability and accuracy including the influence of the geometry approximation.

Discretization Error Analysis of a High Order Unfitted Space-Time Method for moving domain problems

TL;DR

<3-5 sentence high-level summary>This paper analyzes a higher-order unfitted space-time DG method for convection-diffusion problems on moving domains, where the geometry is represented by isoparametric space-time mappings. It extends previous geometric error bounds to the fully discrete setting, including ghost-penalty stabilization and a mass-conserving variant, and establishes stability and coercivity through an inf-sup framework. A Strang-type decomposition yields an a priori error bound that separates geometric, interpolation, and consistency contributions, showing optimal convergence rates under mild geometric regularity and small mesh/time-step parameters. The work is significant for solving moving-domain PDEs without remeshing, enabling high-order accuracy on evolving geometries with robust mass conservation and rigorous error control.

Abstract

We present a numerical analysis of a higher order unfitted space-time Finite Element method applied to a convection-diffusion model problem posed on a moving bulk domain. The method uses isoparametric space-time mappings for the geometry approximation of level set domains and has been presented and investigated computationally in [Heimann, Lehrenfeld, Preuß, SIAM J. Sci. Comp. 45(2), 2023, B139 - B165]. Recently, in [Heimann, Lehrenfeld, IMA J. Numer. Anal., 2025] error bounds for the geometry approximation have been proven. In this paper we prove stability and accuracy including the influence of the geometry approximation.

Paper Structure

This paper contains 22 sections, 26 theorems, 150 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Model problem
  3. Discrete Space-Time geometries and function spaces
  4. The discrete method
  5. Ghost-penalties
  6. Global conservation and a space splitting
  7. Realization of methods
  8. Stability and continuity analysis
  9. General assumptions
  10. Controlling boundary integrals and control by symmetric testing
  11. Ghost-penalty stabilization on deformed space-time meshes
  12. Controlling the time derivative
  13. Stability and Continuity
  14. Strang-type analysis
  15. Geometrical consistency errors
  16. ...and 7 more sections

Key Result

Lemma 3.1

Let $\hat{u}$ be a function defined on $\tilde{\Omega} \times [0,T]$ (or a tensor-product subregion), which is sufficiently smooth s.t. all expressions in this Lemma are well-defined. Let $u := \hat{u} \circ (\Theta_h^{\text{st}})^{-1}$, then for all $S\subseteq \tilde{\Omega}$ with $\mathrm{vol}_d(

Figures (4)

  • Figure 1: Mappings involved in the discrete space-time geometry construction and accuracy analysis.
  • Figure 2: Illustration of an example for $\Omega^{\text{lin}\hbox{!},I_{\hbox{!n}}}_{\mathcal{E}}$and $\Omega^{\text{lin}\hbox{!},I_{\hbox{!n}}}_{\mathcal{I}}$on the undeformed mesh for some time interval $I_n$ with $\partial \Omega^{\text{lin}}(t_{n-1})$and $\partial \Omega^{\text{lin}}(t_{n})$illustrated to indicate the cut topologies.
  • Figure 3: Details of the definition of the jump operator on the curved elements.
  • Figure 4: Illustration of examples of paths from cut elements to the interior as required by \ref{['gpassumption']}. C.f. \ref{['fig:E_I_lin']} for a sketch of the same geometric setting with a highlight on $\Omega^{\text{lin}\hbox{!},I_{\hbox{!n}}}_{\mathcal{E}}$

Theorems & Definitions (57)

  • Lemma 3.1
  • Definition 3.1: Derivative in mesh direction
  • Corollary 3.1
  • Lemma 4.1
  • proof
  • Remark 4.1: Literature on related mass conserving variant methods
  • Remark 5.1: Related stability analyses in the literature
  • Lemma 5.1: Special trace inequality
  • proof
  • Lemma 5.2
  • ...and 47 more