Mixed Finite Element Method and numerical analysis of a convection-diffusion-reaction model in a porous medium

TL;DR

This paper develops a mixed finite element method for a nonlinear convection-diffusion-reaction model in porous media, providing theoretical analysis including existence, uniqueness, stability, and convergence of the numerical scheme.

Contribution

It introduces an explicit flux approach and offers rigorous mathematical analysis for the discretized model's properties.

Findings

Proved existence and uniqueness of the discrete solution

Established stability of the numerical scheme

Demonstrated convergence for semi- and fully discretized models

Abstract

A hydrogeological model for the spread of pollution in an aquifer is considered. The model consists in a convection-diffusion-reaction equation involving the dispersion tensor which depends nonlinearly of the fluid velocity. We introduce an explicit flux in the model and use a mixed Finite Element Method for the discretization. We provide existence, uniqueness and stability results for the discrete model. A convergence result is obtained for the semi-discretized in time problem and for the fully discretization.