A priori and a posteriori error estimates for nonconforming finite element approximations of transport processes in porous media

This work investigates a priori and a posteriori error estimates for finite element approximations of Darcy's equations and convection-diffusion equations in a nonconforming framework. For Darcy's equations, we consider a finite volume box scheme in which the pressure is approximated by the Crouzeix-Raviart finite element and the velocity is then reconstructed locally in the Raviart-Thomas finite element space. For convection-diffusion-reaction equations, we study a Crouzeix-Raviart fnite element approximation stabilized by an edge penalty technique. For both problems, optimal a priori error esti mates are derived. In particular for Darcy's equations, the estimates are uniform in the maximum spatial fluctuations of the hydraulic permeability, a critical property when tackling stongly heterogeneous porous media. A posteriori error estimates are obtained for Darcy's equations using residual and hierarchical techniques. Adaptive quasi-Delaunay type meshes are generated using a frontal method and the error indicators. Numerical results are presented.