Error analysis of a fully discrete Morley finite element approximation for the Cahn-Hilliard equation

This paper proposes and analyzes the Morley element method for the Cahn-Hilliard equation. It is a fourth order nonlinear singular perturbation equation arises from the binary alloy problem in materials science, and its limit is proved to approach the Hele-Shaw flow. If the $L^2(\Omega)$ error estimate is considered directly as in paper \cite{elliott1989nonconforming}, we can only prove that the error bound depends on the exponential function of $\frac{1}{\epsilon}$. Instead, this paper derives the error bound which depends on the polynomial function of $\frac{1}{\epsilon}$ by considering the discrete $H^{-1}$ error estimate first. There are two main difficulties in proving this polynomial dependence of the discrete $H^{-1}$ error estimate. Firstly, it is difficult to prove discrete energy law and discrete stability results due to the complex structure of the bilinear form of the Morley element discretization. This paper overcomes this difficulty by defining four types of discrete inverse Laplace operators and exploring the relations between these discrete inverse Laplace operators and continuous inverse Laplace operator. Each of these operators plays important roles, and their relations are crucial in proving the discrete energy law, discrete stability results and error estimates. Secondly, it is difficult to prove the discrete spectrum estimate in the Morley element space because the Morley element space intersects with the $C^1$ conforming finite element space but they are not contained in each other. Instead of proving this discrete spectrum estimate in the Morley element space, this paper proves a generalized coercivity result by exploring properties of the enriching operators and using the discrete spectrum estimate in its $C^1$ conforming relative finite element space, which can be obtained by using the spectrum estimate of the Cahn-Hilliard operator.