A Posteriori Error Estimates for Nonconforming Finite Element Approximations of an Anisotropic Elliptic Problem

In this paper, we develop an a posteriori error estimator for the $P_1$-nonconforming finite element approximation of a reaction-diffusion equation in two and three dimensions, allowing for data that are not necessarily piecewise constant. Our approach is based on a post-processing technique that constructs a conforming finite element approximation from the nonconforming solution. The a posteriori error estimator is defined as the local norm of the difference between the nonconforming and conforming approximations within each element, augmented by a residual term. Importantly, we provide explicit constants that demonstrate the robustness of the estimator. Finally, we present numerical results illustrating the estimator's performance in the context of an adaptive mesh refinement strategy.