Localized pointwise a posteriori error estimates for nonconforming finite element methods

This paper establishes 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. The key ingredient 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 equation.