Stable Crank–Nicolson discretisation for incompressible miscible displacement problems of low regularity

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.