Superconvergence analysis of a two-grid method for an energy-stable Ciarlet-Raviart type scheme of Cahn-Hilliard equation

In this paper, superconvergence analysis of a mixed finite element method (MFEM) combined with the two-grid method (TGM) is presented for the Cahn-Hilliard (CH) equation for the first time. In particular, the discrete energy-stable Ciarlet-Raviart scheme is constructed with the bilinear element. By use of the high accuracy character of the element, the superclose estimates are deduced for both of the traditional MFEM and of the TGM. Crucially, the main difficulty brought by the coupling of the unknowns is dealt with by some technical methods. Furthermore, the global superconvergent results are achieved by interpolation postprocessing skill. Numerical results illustrate that the proposed TGM is very effective and its computing cost is almost one-third of that of the traditional FEM without loss of accuracy.