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Analysis of a space-time unfitted finite element method for PDEs on evolving surfaces

Arnold Reusken, Hauke Sass

TL;DR

Basic results are derived in which certain geometric characteristics of the exact space-time surface are related to corresponding ones of the numerical surface approximation of the space-time TraceFEM.

Abstract

In this paper we analyze a space-time unfitted finite element method for the discretization of scalar surface partial differential equations on evolving surfaces. For higher order approximations of the evolving surface we use the technique of (iso)parametric mappings for which a level set representation of the evolving surface is essential. We derive basic results in which certain geometric characteristics of the exact space-time surface are related to corresponding ones of the numerical surface approximation. These results are used in an error analysis of a higher order space-time TraceFEM.

Analysis of a space-time unfitted finite element method for PDEs on evolving surfaces

TL;DR

Basic results are derived in which certain geometric characteristics of the exact space-time surface are related to corresponding ones of the numerical surface approximation of the space-time TraceFEM.

Abstract

In this paper we analyze a space-time unfitted finite element method for the discretization of scalar surface partial differential equations on evolving surfaces. For higher order approximations of the evolving surface we use the technique of (iso)parametric mappings for which a level set representation of the evolving surface is essential. We derive basic results in which certain geometric characteristics of the exact space-time surface are related to corresponding ones of the numerical surface approximation. These results are used in an error analysis of a higher order space-time TraceFEM.
Paper Structure (15 sections, 20 theorems, 167 equations, 3 figures)

This paper contains 15 sections, 20 theorems, 167 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Level set representation of the continuous space-time surface
  3. Approximation of the space-time surface
  4. Extension techniques
  5. Basic properties of the space-time surface approximation
  6. Partial integration formula on the space-time surface approximation
  7. Transformation between space-time surface differential operators
  8. Basic geometric error estimates
  9. Application to a surface diffusive transport equation
  10. Space-time trace finite element discretization
  11. Discretization error analysis
  12. Stability estimate and Strang lemma
  13. Consistency error analysis
  14. Interpolation and discretization error bounds
  15. Appendix: proof of Lemma \ref{['lemmeasure']}

Key Result

Lemma 4.1

\newlabel[lemma]lemmeasure Let the three-dimensional surface measures of $S$ and $S_h$ be denoted by $\dif \sigma$ and $\dif \sigma_h$, respectively. Let $\mu_h^S$ be such that ${\mu_h^S\dif \sigma_h=\dif \sigma\circ \mathbf p}$ on $S_h^n$. We define $\mathbf n_0^T\coloneqq\left(\mathbf n^T,0\right

Figures (3)

  • Figure 3.1: The mapping $\Theta^{n}_{h}$, defined on $Q_{h,n}^S$, deforms the surface $S_{\operatorname{lin}}$. Note that $S_{\operatorname{lin}}$ is piecewise linear in space only.
  • Figure 4.1: The conormals $\pm \boldsymbol{\nu}_{\partial}$ at the time slab boundaries and the interior conormals $\boldsymbol{\nu}_{h}\raisebox{-.5ex}{$|$}_{K_S}$. Due to the non-smoothness of $S_h$, in general at a common face $F$ one has $(\boldsymbol{\nu}_{h}\raisebox{-.5ex}{$|$}_{K_S^1})\raisebox{-.5ex}{$|$}_{F}\neq -(\boldsymbol{\nu}_{h}\raisebox{-.5ex}{$|$}_{K_S^2})\raisebox{-.5ex}{$|$}_{F}$.
  • Figure 5.1: Illustration of normals and conormals for two neighbouring elements $K_S^1, K_S^2\in \mathcal{T}_{S_h}$.

Theorems & Definitions (44)

  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Lemma 4.1
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • proof
  • ...and 34 more