Analysis of the Morley element for the Cahn-Hilliard equation and the Hele-Shaw flow

Abstract

The paper analyzes the Morley element method for the Cahn-Hilliard equation. The objective is to prove the numerical interfaces of the Morley element method approximate the Hele-Shaw flow. It is achieved by establishing the optimal error estimates which depend on 1/ϵ polynomially, and the error estimates should be established from lower norms to higher norms progressively. If the higher norm error bound is derived by choosing test function directly, we cannot obtain the optimal error order, and we cannot establish the error bound which depends on 1/ϵ polynomially either. Different from the discontinuous Galerkin (DG) space [Feng et al. SIAM J. Numer. Anal. 54 (2016) 825-847], the Morley element space does not contain the finite element space as a subspace such that the projection theory does not work. The enriching theory is used in this paper to overcome this difficulty, and some nonstandard techniques are combined in the process such as the a priori estimates of the exact solution u, integration by parts in space, summation by parts in time, and special properties of the Morley elements. If one of these techniques is lacked, either we can only obtain the sub-optimal piecewise L∞(H2) error order, or we can merely obtain the error bounds which are exponentially dependent on 1/ϵ. Numerical results are presented to validate the optimal L∞(H2) error order and the asymptotic behavior of the solutions of the Cahn-Hilliard equation.