Analysis of mixed interior penalty discontinuous Galerkin methods for the Cahn-Hilliard equation and the Hele-Shaw flow

Abstract: This paper proposes and analyzes two fully discrete mixed interior penalty discontinuous Galerkin (DG) methods for the fourth order nonlinear Cahn-Hilliard equation. Both methods use the backward Euler method for time discretization and interior penalty discontinuous Galerkin methods for spatial discretization. They differ from each other on how the nonlinear term is treated, one of them is based on fully implicit time-stepping and the other uses the energy-splitting time-stepping. The primary goal of the paper is to prove the convergence of the numerical interfaces of the DG methods to the interface of the Hele-Shaw flow. This is achieved by establishing error estimates that depend on $\epsilon^{-1}$ only in some low polynomial orders, instead of exponential orders. Similar to [14], the crux is to prove a discrete spectrum estimate in the discontinuous Galerkin finite element space. However, the validity of such a result is not obvious because the DG space is not a subspace of the (energy) space $H^1$ and it is larger than the finite element space. This difficult is overcome by a delicate perturbation argument which relies on the discrete spectrum estimate in the finite element space proved in \cite{Feng_Prohl04}. Numerical experiment results are also presented to gauge the theoretical results and the performance of the proposed fully discrete mixed DG methods.