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 $p\geq1$ and backward Euler discretization in time, we show that the order-parameter $c$ is approximated in the broken ${\rm L}^\infty({\rm H}^1)$ norm, with optimal order ${\cal O}(h^p+\tau)$; the associated chemical potential $w=\Phi'(c)-\gamma^2\Delta c$ is shown to be approximated, with optimal order ${\cal O}(h^p+\tau)$ in the broken ${\rm L}^2({\rm H}^1)$ norm. Here $\Phi(c)=\frac{1}{4}(1-c^2)^2$ is a quartic free-energy function and $\gamma>0$ is an interface parameter. Numerical results are presented with polynomials of degree $p=1,2,3$.