Convergence of a decoupled mixed FEM for miscible displacement in interfacial porous media

TL;DR

This paper proves the unconditional optimal convergence of a decoupled mixed finite element method for simulating miscible displacement in porous media with discontinuous properties, improving upon previous methods that required grid ratio restrictions.

Contribution

It introduces an intermediate elliptic interface problem to establish unconditional convergence of the mixed FEM for complex porous media.

Findings

Optimal-order convergence rate achieved without grid ratio restrictions

Method handles discontinuous permeability and porosity across interfaces

Finite element solutions are uniformly regular in subdomains

Abstract

In this paper, we study the stability and convergence of a decoupled and linearized mixed finite element method (FEM) for incompressible miscible displacement in a porous media whose permeability and porosity are discontinuous across some interfaces. We show that the proposed scheme has optimal-order convergence rate unconditionally, without restriction on the grid ratio (between the time-step size and spatial mesh size). Previous works all required certain restrictions on the grid ratio except for the problem with globally smooth permeability and porosity. Our idea is to introduce an intermediate system of elliptic interface problems, whose solution is uniformly regular in each subdomain separated by the interfaces and its finite element solution coincides with the fully discrete solution of the original problem. In order to prove the boundedness of the fully discrete solution, we study the finite element discretization of the intermediate system of elliptic interface problems.