Error Analysis of an Unconditionally Energy Stable Local Discontinuous Galerkin Scheme for the Cahn-Hilliard Equation with Concentration-Dependent Mobility

In this paper, we mainly study the error analysis of an unconditionally energy stable local discontinuous Galerkin (LDG) scheme for the Cahn-Hilliard equation with concentration-dependent mobility. The time discretization is based on the invariant energy quadratization (IEQ) method. The fully discrete scheme leads to a linear algebraic system to solve at each time step. The main difficulty in the error estimates is the lack of control on some jump terms at cell boundaries in the LDG discretization. Special treatments are needed for the initial condition and the non-constant mobility term of the Cahn-Hilliard equation. For the analysis of the non-constant mobility term, we take full advantage of the semi-implicit time-discrete method and bound some numerical variables in L-infinity-norm by the mathematical induction method. The optimal error results are obtained for the fully discrete scheme.