Discontinuous Galerkin Finite Element Approximation of the Cahn–Hilliard Equation with Convection

The paper is concerned with the construction and convergence analysis of a discontinuous Galerkin finite element method for the Cahn–Hilliard equation with convection. Using discontinuous piecewise polynomials of degree p≥1 and backward Euler discretization in time, we show that the order-parameter c is approximated in the broken L^∞( H¹) norm, with optimal order O(hᵖ+τ); the associated chemical potential w=Φ'(c)-γ²Δ c is shown to be approximated, with optimal order O(hᵖ+τ) in the broken L²( H¹) norm. Here Φ(c)=1/4(1-c²)² is a quartic free-energy function and γ>0 is an interface parameter. Numerical results are presented with polynomials of degree $p=1,2,3$.