Direct solver for the Cahn-Hilliard equation by Legendre-Galerkin spectral method

We propose an efficient algorithm based on the Legendre-Galerkin approximation for the direct solution of the Cahn-Hilliard equation with homogeneous and nonhomogeneous boundary conditions. The fully discretized scheme combines a large-time step splitting method in time and spectral method in space. It is proven that the first order fully discrete numerical solution preserves the energy dissipation property which is also contented in the associated continuous problem. A rigorous error estimate is carried out to establish the convergence rate with respect to time step and the polynomial degree of the method. The numerical results conform the accuracy and the efficiency of the direct solver. Finally, the proposed schemes are applied to the phase field simulation. (C) 2019 Elsevier B.V. All rights reserved.