A Conservative Flux Optimization Finite Element Method for Convection-Diffusion Equations

TL;DR

This paper introduces a finite element method that enhances flux approximation accuracy and local mass conservation for convection-diffusion equations, demonstrating optimal convergence and effective performance in complex flow simulations.

Contribution

The paper develops a conservative flux optimization finite element scheme that ensures local mass conservation and provides convergent approximations for both primal and flux variables.

Findings

Optimal convergence in discrete Sobolev norms

Accurate flux approximation with local mass conservation

Effective in simulating two-phase flow in heterogeneous media

Abstract

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.