New analysis of Galerkin-mixed FEMs for incompressible miscible flow in porous media

TL;DR

This paper provides new, optimal error estimates for Galerkin-mixed finite element methods applied to incompressible miscible flow in porous media, improving understanding of their accuracy and applicability.

Contribution

It establishes unconditionally second-order $L^2$-norm error estimates for the lowest-order Galerkin-mixed FEM, extending the analysis to general cases and confirming results with numerical experiments.

Findings

Unconditional second-order $L^2$-norm accuracy for concentration.

Optimal error estimates for all components of the system.

Numerical validation in 2D and 3D models.

Abstract

Analysis of Galerkin-mixed FEMs for incompressible miscible flow in porous media has been investigated extensively in the last several decades. Of particular interest in practical applications is the lowest-order Galerkin-mixed method, { in which a linear Lagrange FE approximation is used for the concentration and the lowest-order Raviart-Thomas FE approximation is used for the velocity/pressure. The previous works only showed the first-order accuracy of the method in $L^2$-norm in spatial direction,} which however is not optimal and valid only under certain extra restrictions on both time step and spatial mesh. In this paper, we provide new and optimal $L^2$-norm error estimates of Galerkin-mixed FEMs for all three components in a general case. In particular, for the lowest-order Galerkin-mixed FEM, we show unconditionally the second-order { accuracy in $L^2$-norm} for the concentration. Numerical results for both two and three-dimensional models are presented to confirm our theoretical analysis. More important is that our approach can be extended to the analysis of mixed FEMs for many strongly coupled systems to obtain optimal error estimates for all components.