A priori Error Analysis of a Discontinuous Galerkin Method for Cahn–Hilliard–Navier–Stokes Equations

Abstract

In this paper, we analyze an interior penalty discontinuous Galerkin method for solving the coupled Cahn–Hilliard and Navier–Stokes equations. We prove unconditional unique solvability of the discrete system, and we derive stability bounds without any restrictions on the chemical energy density function. The numerical solutions satisfy a discrete energy dissipation law and mass conservation laws. Convergence of the method is obtained by obtaining optimal a priori error estimates.