A Conservative Flux Optimization Finite Element Method for Convection-Diffusion Equations
This work addresses the need for locally mass-conservative flux approximations in finite element methods for convection-diffusion problems, which is important for accurate simulations in porous media flow.
The authors propose a new finite element method for convection-diffusion equations that enforces local mass conservation by optimizing flux approximations. The method achieves optimal convergence rates in discrete Sobolev norms and demonstrates excellent performance in simulations of two-phase flow in heterogeneous porous media.
This article presents a new finite element method for convection-diffusion equations by enhancing the continuous finite element space with a flux space for flux approximations that preserve the important mass conservation locally on each element. The numerical scheme is based on a constrained flux optimization approach where the constraint was given by local mass conservation equations and the flux error is minimized in a prescribed topology/metric. This new scheme provides numerical approximations for both the primal and the flux variables. It is shown that the numerical approximations for the primal and the flux variables are convergent with optimal order in some discrete Sobolev norms. Numerical experiments are conducted to confirm the convergence theory. Furthermore, the new scheme was employed in the computational simulation of a simplified two-phase flow problem in highly heterogeneous porous media. The numerical results illustrate an excellent performance of the method in scientific computing.