Stability and Convergence of a Second Order Mixed Finite Element Method for the Cahn-Hilliard Equation

TL;DR

This paper introduces and analyzes an unconditionally stable, second-order numerical scheme for the Cahn-Hilliard equation, demonstrating its stability, boundedness, and optimal convergence rates in 2D and 3D.

Contribution

The paper presents a novel second-order, unconditionally stable finite element method for the Cahn-Hilliard equation with rigorous stability and convergence analysis.

Findings

Unconditional energy stability proven for the scheme

Discrete variables are bounded in relevant norms

Optimal convergence rates established in energy norms

Abstract

In this paper we devise and analyze an unconditionally stable, second-order-in-time numerical scheme for the Cahn-Hilliard equation in two and three space dimensions. We prove that our two-step scheme is unconditionally energy stable and unconditionally uniquely solvable. Furthermore, we show that the discrete phase variable is bounded in $L^\infty (0,T;L^\infty)$ and the discrete chemical potential is bounded in $L^\infty (0,T;L^2)$, for any time and space step sizes, in two and three dimensions, and for any finite final time $T$. We subsequently prove that these variables converge with optimal rates in the appropriate energy norms in both two and three dimensions. We include in this work a detailed analysis of the initialization of the two-step scheme.