Unconditional convergence and optimal error estimates of a Galerkin-mixed FEM for incompressible miscible flow in porous media

TL;DR

This paper proves that a Galerkin-mixed finite element method for incompressible miscible flow achieves optimal error estimates without requiring time-step restrictions, advancing understanding of linearized schemes for nonlinear parabolic equations.

Contribution

It establishes unconditional convergence and optimal error bounds for a Galerkin-mixed FEM with semi-implicit Euler scheme, removing previous time-step constraints.

Findings

Optimal $L^2$ error estimates without time-step restrictions

Theoretical framework applicable to general nonlinear parabolic systems

Error analysis based on splitting into discretization and finite element errors

Abstract

In this paper, we study the unconditional convergence and error estimates of a Galerkin-mixed FEM with the linearized semi-implicit Euler time-discrete scheme for the equations of incompressible miscible flow in porous media. We prove that the optimal $L^2$ error estimates hold without any time-step (convergence) condition, while all previous works require certain time-step condition. Our theoretical results provide a new understanding on commonly-used linearized schemes for nonlinear parabolic equations. The proof is based on a splitting of the error function into two parts: the error from the time discretization of the PDEs and the error from the finite element discretization of corresponding time-discrete PDEs. The approach used in this paper is applicable for more general nonlinear parabolic systems and many other linearized (semi)-implicit time discretizations.