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Weak maximum principle of finite element methods for parabolic equations in polygonal domains

Genming Bai, Dmitriy Leykekhman, Buyang Li

TL;DR

This work establishes a weak maximum principle for both semi-discrete and fully discrete finite element methods solving the parabolic equation $\partial_t u-\Delta u=0$ on nonsmooth polygonal domains. It develops a dyadic space-time decomposition and leverages discrete Laplace transform techniques together with regularized/ discrete Green's functions to obtain uniform $L^\infty$-bounds that depend on boundary data and initial data, for BDF time-stepping with $k=1$–$6$. The analysis addresses the challenges posed by domain nonsmoothness and time discretization, providing sharp gradient-error estimates for the parabolic Green's function approximation and proving the weak maximum principle under realistic FEM assumptions. It also discusses how the methodology extends to dG time-stepping and highlights limitations related to variable time steps and $A$-stability constraints of higher-order BDF schemes.

Abstract

The weak maximum principle of finite element methods for parabolic equations is proved for both semi-discretization in space and fully discrete methods with $k$-step backward differentiation formulae for $k = 1,... ,6$, on a two-dimensional general polygonal domain or a three-dimensional convex polyhedral domain. The semi-discrete result is established via a dyadic decomposition argument and local energy estimates in which the nonsmoothness of the domain can be handled. The fully discrete result for multistep backward differentiation formulae is proved by utilizing the solution representation via the discrete Laplace transform and the resolvent estimates, which are inspired by the analysis of convolutional quadrature for parabolic and fractional-order partial differential equations.

Weak maximum principle of finite element methods for parabolic equations in polygonal domains

TL;DR

This work establishes a weak maximum principle for both semi-discrete and fully discrete finite element methods solving the parabolic equation on nonsmooth polygonal domains. It develops a dyadic space-time decomposition and leverages discrete Laplace transform techniques together with regularized/ discrete Green's functions to obtain uniform -bounds that depend on boundary data and initial data, for BDF time-stepping with –. The analysis addresses the challenges posed by domain nonsmoothness and time discretization, providing sharp gradient-error estimates for the parabolic Green's function approximation and proving the weak maximum principle under realistic FEM assumptions. It also discusses how the methodology extends to dG time-stepping and highlights limitations related to variable time steps and -stability constraints of higher-order BDF schemes.

Abstract

The weak maximum principle of finite element methods for parabolic equations is proved for both semi-discretization in space and fully discrete methods with -step backward differentiation formulae for , on a two-dimensional general polygonal domain or a three-dimensional convex polyhedral domain. The semi-discrete result is established via a dyadic decomposition argument and local energy estimates in which the nonsmoothness of the domain can be handled. The fully discrete result for multistep backward differentiation formulae is proved by utilizing the solution representation via the discrete Laplace transform and the resolvent estimates, which are inspired by the analysis of convolutional quadrature for parabolic and fractional-order partial differential equations.
Paper Structure (16 sections, 13 theorems, 195 equations)

This paper contains 16 sections, 13 theorems, 195 equations.

Table of Contents

  1. Introduction
  2. The main results
  3. Notation, finite element discretization, and regularized Green's function
  4. Function spaces and notation
  5. Properties of the finite element spaces
  6. Green's functions
  7. Weak maximum principle of semi-discrete FEM
  8. Dyadic decomposition of the domain ${\mathcal{Q}}=(0,1)\times\Omega$
  9. Technical lemmas
  10. Reduction of the problem
  11. Proof of \ref{['grad-Fh-estimate']}
  12. Weak maximum principle of fully discrete FEM
  13. Proof of \ref{['uN-Linft-proved']}
  14. Proof of \ref{['BDF-FEM-f']}
  15. Proof of \ref{['Eq-uh3-Linfty']}
  16. ...and 1 more sections

Key Result

theorem 2.1

If $\Omega$ is a polygon (possibly nonconvex) in $\mathbb{R}^2$ or a convex polyhedron in $\mathbb{R}^3$, then the solution of the semi-discrete FEM in semi-discrete-FEM satisfies the following weak maximum principle for $r\ge 1$ in $\mathbb{R}^2$ and $r\ge 2$ in $\mathbb{R}^3$: where $C$ is a constant independent of $h$ and $T$.

Theorems & Definitions (19)

  • theorem 2.1: Weak maximum principle of semi-discrete FEM
  • theorem 2.2: Weak maximum principle of fully discrete FEM
  • remark 2.1
  • lemma 4.1
  • lemma 4.2
  • lemma 4.3
  • proof
  • lemma 4.4
  • lemma 5.1
  • proof
  • ...and 9 more