Localized pointwise a posteriori error estimates for nonconforming finite element methods

Abstract

This paper establishes the first localized pointwise a posteriori error estimates for nonconforming finite element discretizations of the Poisson and biharmonic equations. For the Poisson problem, we derive localized estimates for the function-value and broken gradient errors of the Crouzeix--Raviart method. For the Morley discretization of the biharmonic equation, we derive a localized a posteriori estimate that controls the local Hessian error. This provides the first pointwise a posteriori error analysis for the biharmonic equation, for either conforming or nonconforming finite element methods. The key ingredient, absent from pointwise analysis of second order PDEs, is the design of two novel weight functions that facilitate sharp estimates of the $L^1$ norms of derivatives of a regularized Green's function for the biharmonic operator.