A reduced-order finite element method based on POD for the incompressible miscible displacement problem

Abstract

In this paper, we establish a reduced-order finite element (ROFE) method with very few degrees of freedom for the incompressible miscible displacement problem. Firstly, we construct the finite element (FE) method with second-order accuracy in time, where backward difference formulation is used for the time discretization and classical FE formulation is used for the space discretization. Optimal a priori error estimates for the FE solutions are proved. Secondly, we apply the proper orthogonal decomposition (POD) technique to develop the ROFE method, which can effectively reduce degrees of freedom and CPU time. Optimal a priori error estimates for the ROFE solutions are derived. Besides, we give the algorithm process of the ROFE method. Finally, some numerical examples are presented to verify the behavior of the ROFE method for piecewise linear element. And these examples imply the proposed method is feasible and effective for solving the incompressible miscible displacement problem.