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)−γ2∆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.