A posteriori error estimation based on conservative flux reconstruction for nonconforming finite element approximations to a singularly perturbed reaction–diffusion problem on anisotropic meshes

Abstract

Based on conservative flux reconstruction, we derive a posteriori error estimates for the nonconforming finite element approximations to a singularly perturbed reaction–diffusion problem on anisotropic meshes, since the solution exhibits boundary or interior layers and this anisotropy is reflected in a discretization by using anisotropic meshes. Without the assumption that the meshes are shape-regular, our estimates give the upper bounds on the error only containing the alignment measure factor, and therefore, could provide actual numerical bounds if the alignment measure was approximated very well. Two resulting error estimators are presented to be equivalent to each other up to a data oscillation, one of which can be directly constructed without solving local Neumann problems and provide computable error bound. Numerical experiments confirm that our estimates are reliable and efficient as long as the singularly perturbed problem is discretized by a suitable mesh which leads to a small alignment measure.