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Inf-Sup Stability of Parabolic TraceFEM

Lucas Bouck, Ricardo H. Nochetto, Mansur Shakipov, Vladimir Yushutin

Abstract

We develop a parabolic inf-sup theory for a modified TraceFEM semi-discretization in space of the heat equation posed on a stationary surface embedded in $\mathbb{R}^n$. We consider the normal derivative volume stabilization and add an $L^2$-type stabilization to the time derivative. We assume that the representation of and the integration over the surface are exact, however, all our results are independent of how the surface cuts the bulk mesh. For any mesh for which the method is well-defined, we establish necessary and sufficient conditions for inf-sup stability of the proposed TraceFEM in terms of $H^1$-stability of a stabilized $L^2$-projection and of an inverse inequality constant that accounts for the lack of conformity of TraceFEM. Furthermore, we prove that the latter two quantities are bounded uniformly for a sequence of shape-regular and quasi-uniform bulk meshes. We derive several consequences of uniform discrete inf-sup stability, namely uniform well-posedness, discrete maximal parabolic regularity, parabolic quasi-best approximation, convergence to minimal regularity solutions, and optimal order-regularity energy and $L^2 L^2$ error estimates. We show that the additional stabilization of the time derivative restores optimal conditioning of time-discrete TraceFEM typical of fitted discretizations.

Inf-Sup Stability of Parabolic TraceFEM

Abstract

We develop a parabolic inf-sup theory for a modified TraceFEM semi-discretization in space of the heat equation posed on a stationary surface embedded in . We consider the normal derivative volume stabilization and add an -type stabilization to the time derivative. We assume that the representation of and the integration over the surface are exact, however, all our results are independent of how the surface cuts the bulk mesh. For any mesh for which the method is well-defined, we establish necessary and sufficient conditions for inf-sup stability of the proposed TraceFEM in terms of -stability of a stabilized -projection and of an inverse inequality constant that accounts for the lack of conformity of TraceFEM. Furthermore, we prove that the latter two quantities are bounded uniformly for a sequence of shape-regular and quasi-uniform bulk meshes. We derive several consequences of uniform discrete inf-sup stability, namely uniform well-posedness, discrete maximal parabolic regularity, parabolic quasi-best approximation, convergence to minimal regularity solutions, and optimal order-regularity energy and error estimates. We show that the additional stabilization of the time derivative restores optimal conditioning of time-discrete TraceFEM typical of fitted discretizations.
Paper Structure (35 sections, 43 theorems, 236 equations, 1 figure)

This paper contains 35 sections, 43 theorems, 236 equations, 1 figure.

Table of Contents

  1. Introduction
  2. TraceFEM
  3. Motivation
  4. The surface heat equation
  5. Stabilized L2-projection
  6. Our main results
  7. Preliminaries
  8. Bulk mesh
  9. Stabilization of TraceFEM
  10. Inverse parameter
  11. Symmetry relation for Ph
  12. Relation between the dual norms
  13. Inf-sup stability analysis
  14. Continuous setting
  15. Semidiscretization with TraceFEM
  16. ...and 20 more sections

Key Result

Lemma 2.6

Let $\mathcal{T}_h$ satisfy Assumption assump:res-of-geom (resolution of the geometry). If $\|P_h\|_{\mathcal{L}(H^1_*)}$ is the operator norm eq:intro:operator_norms of the stabilized $L^2$-projection eq:intro:l2proj and ${C_{\mathrm{inv}, h}}$ is given by Definition def:Cinvh (inverse parameter),

Figures (1)

  • Figure 1: Asymptotic behavior as $\Delta t \to 0$ of the condition numbers $\kappa(\mathbf B_*), \kappa(\mathbf B)$ for a fixed $h=1/32$, , where $\mathbf B_*$ is given in \ref{['eq:matrices']} and $\mathbf B$ in \ref{['eq:cond-B']}. Note that $\kappa(\mathbf B)$ grows as $(\Delta t)^{-1}$ whereas $\kappa(\mathbf B_*)$ is about constant with respect to $\Delta t$.

Theorems & Definitions (87)

  • Definition 2.2: normal derivative volume stabilization
  • Definition 2.3: stabilized norms
  • Definition 2.4: discrete dual norm
  • Definition 2.5: inverse parameter
  • Lemma 2.6: relation between the dual norms
  • proof
  • Proposition 3.1: well-posedness of the continuous problem
  • proof
  • Theorem 3.2: $V_h$-dependent inf-sup stability
  • Lemma 3.3: stability of auxiliary problem
  • ...and 77 more