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

Abstract

In this paper, we consider the finite element approximation for a parabolic problem on a smooth domain $Ω\subset \mathbb{R}^N$ 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 $\{ Ω_h \}_h$ that approximate the original domain $Ω$. 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 $Ω\bigtriangleup Ω_h$ 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 $\partialΩ$. 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.