Numerical analysis of a discontinuous Galerkin method for Cahn-Hilliard-Navier-Stokes equations

TL;DR

This paper provides a comprehensive theoretical analysis of an interior penalty discontinuous Galerkin method for the coupled Cahn-Hilliard-Navier-Stokes equations, establishing stability, energy dissipation, and convergence properties.

Contribution

It introduces a rigorous analysis demonstrating unconditional solvability, energy stability, and optimal error estimates for the DG method applied to this complex multiphysics problem.

Findings

Unconditional unique solvability of the discrete system.

Establishment of an unconditional discrete energy dissipation law.

Optimal a priori error estimates demonstrating convergence.

Abstract

In this paper, we derive a theoretical analysis of an interior penalty discontinuous Galerkin methods for solving the Cahn-Hilliard-Navier-Stokes model problem. We prove unconditional unique solvability of the discrete system, obtain unconditional discrete energy dissipation law, and derive stability bounds with a generalized chemical energy density. Convergence of the method is obtained by proving optimal a priori error estimates. Our analysis of the unique solvability is valid for both symmetric and non-symmetric versions of the discontinuous Galerkin formulation.