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 $\Omega \subset {\mathbb{R}}^N$ 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 $\Omega △{\Omega }_h$ 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.

31 pages, 1 figure

References in corpus (2)

• Pointwise best approximation results for Galerkin finite element solutions of parabolic problems
• $L^{\infty }$- and $W^{1,\infty }$-error estimates of linear finite element method for Neumann boundary value problems in a smooth domain

Cited by in corpus (2)

• Weak discrete maximum principle of finite element methods in convex polyhedra
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