Maximum norm error estimates for the finite element approximation of parabolic problems on smooth domains

TL;DR

This paper derives $L^ abla$-error estimates for finite element approximations of parabolic problems on smooth, possibly non-convex domains, addressing domain approximation effects and introducing new analytical techniques.

Contribution

It provides new $L^ abla$-error estimates for finite element methods on smooth domains with non-convexities, considering domain approximation errors and boundary effects.

Findings

Established $L^ abla$-error estimates for finite element approximations.

Analyzed the impact of domain approximation and symmetric difference on convergence.

Presented smoothing properties and maximal regularity results as corollaries.

Abstract

In this paper, we consider the finite element approximation for a parabolic problem on a smooth domain $\Omega \subset \mathbb{R}^N$ with the inhomogeneous Neumann boundary condition. We emphasize that the domain can be non-convex in general. We implement the finite element method for this problem by constructing a family of polygonal or polyhedral domains $\{ \Omega_h \}_h$ that approximate the original domain $\Omega$. The main result of this study is the $L^\infty$-error estimate for this approximation. We shall show that the convergence rate is not optimal for higher order elements since the symmetric difference $\Omega \bigtriangleup \Omega_h$ is not empty in general. In order to address the effect of the symmetric difference of domains, we introduce the tubular neighborhood of the original boundary $\partial\Omega$. We will also present a slightly new approach to establish the $L^\infty$-error estimate. Moreover, we present the smoothing property for the discrete parabolic semigroup and the spatially discretized maximal regularity as corollaries of the main result.