New residual-based a posteriori error estimators for lowest-order Raviart-Thomas element approximation to convection-diffusion-reaction equations

Abstract

A new technique of residual-type a posteriori error analysis is developed for the lowest-order Raviart-Thomas mixed finite element discretizations of convection-diffusion-reaction equations in two- or three-dimension. Both centered mixed scheme and upwind-weighted mixed scheme are considered. The a posteriori error estimators, derived for the stress variable error plus scalar displacement error in $L^{2}$-norm, can be directly computed with the solutions of the mixed schemes without any additional cost, and are robust with respect to the coefficients in the equations. Local efficiency dependent on local variations in coefficients is obtained without any saturation assumption, and holds from the cases where convection or reaction is not present to convection- or reaction-dominated problems. The main tools of analysis are the postprocessed approximation of scalar displacement, abstract error estimates, and the property of modified Oswald interpolation. Numerical experiments are reported to support our theoretical results and to show the competitive behavior of the proposed posteriori error estimates.