Robust a posteriori error estimates for conforming and nonconforming finite element methods for convection–diffusion problems

Abstract

A posteriori error estimation is carried out within a unified framework for various conforming and nonconforming finite element methods for convection–diffusion problems. Our main contribution is finding an appropriate norm to measure the error, which incorporates a discrete energy norm, a discrete dual semi-norm of the convective derivative and jumps of the approximate solution over element faces (edges in two dimensions). The error estimator is shown to be robust with respect to the Péclet number in the sense of the modified norm. Based on a general error decomposition, we show that the key ingredient of error estimation is the estimation on the consistency error related to the particular numerical scheme, and the remaining terms can be bounded in a unified way. The numerical results are presented to illustrate the robustness and practical performance of the estimator in an adaptive refinement strategy.