A new optimal error analysis of a mixed finite element method for advection-diffusion-reaction Brinkman flow

This article deals with the error analysis of a Galerkin-mixed finite element methods for the advection-reaction-diffusion Brinkman flow in porous media. Numerical methods for incompressible miscible flow in porous media have been studied extensively in the last several decades. In practical applications, the lowest-order Galerkin-mixed method is the most popular one, where the linear Lagrange element is used for the concentration and the lowest-order Raviart-Thomas element, the lowest-order Nedelec edge element and piece-wise constant discontinuous Galerkin element are used for the velocity, vorticity and pressure, respectively. The existing error estimate of this lowest-order finite element method is only O(h)$$ O(h) $$ for all variables in spatial direction, which is not optimal for the concentration variable. This paper focuses on a new and optimal error estimate of a linearized backward Euler Galerkin-mixed FEMs, where the second-order accuracy for the concentration in spatial directions is established unconditionally. The key to our optimal error analysis is a new negative norm estimate for Nedelec edge element. Moreover, based on the computed numerical concentration, we propose a simple one-step recovery technique to obtain a new numerical velocity, vorticity and pressure with second-order accuracy. Numerical experiments are provided to confirm our theoretical analysis.