ON THE SUPERCONVERGENCE OF A HYBRIDIZABLE DISCONTINUOUS GALERKIN METHOD FOR THE CAHN-HILLIARD EQUATION

We propose a hybridizable discontinuous Galerkin (HDG) method with the convex-concave splitting temporal discretization for solving the Cahn-Hilliard equation. We establish optimal convergence rates for the scalar variables and the flux variables in the L² norm for polynomials of degree k ≥ 0. The error constants depend on inverse of the interface thickness in polynomial orders, which is obtained by utilizing a spectral-type estimate of the discrete Cahn-Hilliard operator in the HDG framework. In terms of degrees of freedom of the globally coupled unknowns, the scalar variables are superconvergent. Numerical results are reported to corroborate the theoretical convergence rates and the effectiveness of the method.