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 derive the optimal error estimates and to prove the zero-level sets of the Cahn-Hilliard equation approximate the Hele-Shaw flow. If the piecewise $L^{\infty}(H^2)$ 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 $\frac{1}ε$ polynomially either. To overcome this difficulty, this paper proves them by the following steps, and the result in each next step cannot be established without using the result in its previous one. First, it proves some a priori estimates of the exact solution $u$, and these regularity results are minimal to get the main results; Second, it establishes ${L^{\infty}(L^2)}$ and piecewise ${L^2(H^2)}$ error bounds which depend on $\frac{1}ε$ polynomially based on the piecewise ${L^{\infty}(H^{-1})}$ and ${L^2(H^1)}$ error bounds; Third, it establishes piecewise ${L^{\infty}(H^2)}$ optimal error bound which depends on $\frac{1}ε$ polynomially based on the piecewise ${L^{\infty}(L^2)}$ and ${L^2(H^2)}$ error bounds; Finally, it proves the ${L^\infty(L^\infty)}$ error bound and the approximation to the Hele-Shaw flow based on the piecewise ${L^{\infty}(H^2)}$ error bound. The nonstandard techniques are used in these steps such as the generalized coercivity result, integration by part in space, summation by part 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 ${L^{\infty}(H^2)}$ error order, or we can merely obtain the error bounds which are exponentially dependent on $\frac{1}ε$. Numerical results are presented to validate the optimal $L^\infty(H^2)$ error order and the asymptotic behavior of the solutions of the Cahn-Hilliard equation.