A priori error estimates for a semi‐Lagrangian unified finite element method for coupled Darcy‐transport problems

Abstract

A semi‐Lagrangian unified finite element method is investigated for solving time‐dependent coupled Darcy‐transport problems. In this method, the modified method of characteristics is combined with a Galerkin finite element discretization allowing the same finite element space to be used for all solutions of the problem including the pressure, velocity and concentration. Convergence and stability of the method are also analyzed in this study and error estimates in the ‐norm are established for the numerical solutions. For the time integration we consider a second‐order implicit method and discrete error estimates are provided. Due to the Lagrangian treatment of convection terms in the considered problems, the conventional Courant‐Friedrichs‐Lewy condition is relaxed and the time truncation errors are reduced in the diffusion‐reaction part. Numerical results are presented for several test examples to demonstrate the reliability and efficiency of the proposed method. The computed results support our expectations for an accurate and highly efficient semi‐Lagrangian unified finite element method for time‐dependent coupled Darcy‐transport problems.