Stable Crank-Nicolson Discretisation for Incompressible Miscible Displacement Problems of Low Regularity

TL;DR

This paper presents a convergent numerical method combining Crank-Nicolson, mixed finite elements, and discontinuous Galerkin techniques for incompressible miscible displacement problems, effective even with low regularity.

Contribution

It introduces a novel convergence proof for low regularity scenarios and demonstrates second-order accuracy and robustness through numerical experiments.

Findings

Second-order convergence for smooth problems

Robustness for rough problems

Effective numerical approximation under low regularity

Abstract

In this article we study the numerical approximation of incompressible miscible displacement problems with a linearised Crank-Nicolson time discretisation, combined with a mixed finite element and discontinuous Galerkin method. At the heart of the analysis is the proof of convergence under low regularity requirements. Numerical experiments demonstrate that the proposed method exhibits second-order convergence for smooth and robustness for rough problems.