Localized pointwise a posteriori error estimates for gradients of piecewise linear finite element approximations to second-order quasilinear elliptic problems

Abstract

Abstract. Two types of pointwise a posteriori error estimates are presented for gradients of finite element approximations of second-order quasilinear elliptic Dirichlet boundary value problems over convex polyhedral domains Ω in space dimension n ≥ 2. We first give a residual estimator which is equivalent to ∇(u − u h ) L∞(Ω) up to higher-order terms. The second type of residual estimator is designed to control ∇(u − u h ) locally over any subdomain of Ω. It is a novel a posteriori counterpart to the localized or weighted a priori estimates of [Sch98]. This estimator is shown to be equivalent (up to higher-order terms) to the error measured in a weighted global norm which depends on the subdomain of interest. All estimates are proved for general shape-regular meshes which may be highly graded and unstructured. The constants in the estimates depend on the unknown solution u in the nonlinear case, but in a fashion which places minimal restrictions on the regularity of u.Key words. finite element methods, quasilinear elliptic problems, a posteriori error estimation, pointwise error analysis

Local pointwise a posteriori gradient error bounds for the Stokes equations

“…Note also that the techniques we employ here could be easily used to extend the global W 1 ∞ a posteriori estimates of [25] to include convex polyhedral domains with no restriction on the maximum interior dihedral angle. In the context of Poisson's problem, we refer to [4] for a posteriori estimates and to [11] for a priori estimates which similarly use sharp Green's function estimates to obtain pointwise gradient bounds on any convex polyhedral domain.…”

“…Local a posteriori estimates for maximum gradient errors for Poisson's problem are proved in [5]; related global maximum gradient error estimators are developed in [4]. In both of these works, the regularization penalty in (1.6) was essentially assumed a priori to be small.…”

Finite volume element method for monotone nonlinear elliptic problems

“…In this article, we focus our attention on posing the problem on polygonal domain [34, 26], even though theoretical results are not always available on polygonal domains. Obviously, smooth boundaries are important for many nonlinear problems.…”

Hölder estimates for Green’s functions on convex polyhedral domains and their applications to finite element methods

“…[6,7,9,11,29]). Many important contributions have been made in order to establish (1.2) with various assumptions on the finite element spaces and geometry of Ω.…”

“…[4,16]), a posteriori residual type estimators (cf. [6]), localized pointwise error estimates for quasilinear problems (cf. [8]), and Richardson Extrapolation (cf.…”