Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Maximal regularity of evolving FEMs for parabolic equations on an evolving surface

Genming Bai, Balázs Kovács, Buyang Li

TL;DR

The work addresses the gap in understanding spatially discrete maximal $L^p$-regularity for parabolic equations on evolving surfaces by establishing it for semi-discrete isoparametric FEMs on evolving hypersurfaces. The authors first develop a stationary-surface theory using Green's functions, dyadic localization, and local energy estimates to prove discrete maximal regularity with mesh-independent constants, including analytic and maximum-norm stability of the discrete heat semigroup. They then extend these results to evolving surfaces via a perturbation argument in time, introducing time-dependent pull-back coefficients and a continuous surrogate for the discrete operator, and applying Grönwall-type arguments to preserve the maximal-regularity bounds. The results enable robust, optimal-order $L^p$-norm error analysis for nonlinear and quasi-linear parabolic PDEs on evolving surfaces and provide a rigorous foundation for numerical simulations in geometric evolutions and surface diffusion problems.

Abstract

In this paper, we prove that spatially semi-discrete evolving finite element method for parabolic equations on a given evolving hypersurface of arbitrary dimensions preserves the maximal $L^p$-regularity at the discrete level. We first establish the results on a stationary surface and then extend them, via a perturbation argument, to the case where the underlying surface is evolving under a prescribed velocity field. The proof combines techniques in evolving finite element method, properties of Green's functions on (discretised) closed surfaces, and local energy estimates for finite element methods

Maximal regularity of evolving FEMs for parabolic equations on an evolving surface

TL;DR

The work addresses the gap in understanding spatially discrete maximal -regularity for parabolic equations on evolving surfaces by establishing it for semi-discrete isoparametric FEMs on evolving hypersurfaces. The authors first develop a stationary-surface theory using Green's functions, dyadic localization, and local energy estimates to prove discrete maximal regularity with mesh-independent constants, including analytic and maximum-norm stability of the discrete heat semigroup. They then extend these results to evolving surfaces via a perturbation argument in time, introducing time-dependent pull-back coefficients and a continuous surrogate for the discrete operator, and applying Grönwall-type arguments to preserve the maximal-regularity bounds. The results enable robust, optimal-order -norm error analysis for nonlinear and quasi-linear parabolic PDEs on evolving surfaces and provide a rigorous foundation for numerical simulations in geometric evolutions and surface diffusion problems.

Abstract

In this paper, we prove that spatially semi-discrete evolving finite element method for parabolic equations on a given evolving hypersurface of arbitrary dimensions preserves the maximal -regularity at the discrete level. We first establish the results on a stationary surface and then extend them, via a perturbation argument, to the case where the underlying surface is evolving under a prescribed velocity field. The proof combines techniques in evolving finite element method, properties of Green's functions on (discretised) closed surfaces, and local energy estimates for finite element methods
Paper Structure (17 sections, 13 theorems, 159 equations)

This paper contains 17 sections, 13 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Evolving surface FEMs and main results
  3. Parabolic equations on an evolving surface
  4. Evolving surface finite elements
  5. Lift
  6. The semi-discrete problems
  7. Main results: Spatially discrete maximal regularity
  8. Preliminary results on stationary surfaces
  9. Function spaces
  10. Properties of the finite element space
  11. Green's functions
  12. $W^{1,q}$-stability of Ritz projection
  13. Dyadic decomposition of the domain ${\mathcal{Q}}=(0,1)\times\Gamma$
  14. Local energy estimates
  15. Discrete maximal regularity on a stationary surface (Proof of Theorem \ref{['thm:disc_max_reg2']})
  16. ...and 2 more sections

Key Result

Theorem 2.1

If $u_0 = 0$, then the solution $U$ of the pulled-back PDE eq:PDE_pb obeys the following estimate for $p,q\in(1,\infty)$: where the constant $C>0$ only depends on $\mathcal{G}_T$.

Theorems & Definitions (19)

  • Theorem 2.1: Maximal regularity of continuous PDE, KL23
  • Theorem 2.2: Maximal regularity of semi-discrete surface FEMs on a stationary surface
  • Theorem 2.3: Maximal regularity of semi-discrete surface FEM on an evolving surface
  • Remark 3.1
  • Lemma 3.1
  • Lemma 3.2
  • Remark 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Remark 3.3
  • ...and 9 more