Hessian recovery based finite element methods for the Cahn-Hilliard Equation

TL;DR

This paper introduces a novel finite element method for the Cahn-Hilliard equation that uses a recovery-based approach to discretize the fourth-order operator within a standard linear finite element space, combining finite difference and finite element techniques.

Contribution

The paper presents a new recovery-based finite element method that discretizes the Cahn-Hilliard equation's fourth-order operator using a least-squares Hessian recovery, linking it to finite difference schemes.

Findings

Optimal-order convergence demonstrated.

Energy stability verified numerically.

Laplace recovery scheme matches five-point stencil.

Abstract

In this paper, we propose a novel recovery based finite element method for the Cahn-Hilliard equation. One distinguishing feature of the method is that we discretize the fourth-order differential operator in a standard $C^0$ linear finite elements space. Precisely, we first transform the fourth-order Cahn-Hilliard equation to its variational formulation in which only first-order and second-order derivatives are involved and then we compute the and second-order derivatives of a linear finite element function by a least-squares fitting recovery procedure. The intrinsic link between the second-order derivatives (Hessian matrix) recovery scheme and the finite difference method is studied in the paper. In particular, for the first time, we discover that the Laplace recovery scheme is exactly the well-known five-point stencil over uniform meshes. The proposed discretization for the Cahn-Hilliard equation can be regarded as a combination of the finite difference scheme and the finite element scheme. In addition, special considerations are put on different methods for imposing Neumann type boundary conditions. The optimal-order convergence and energy stability are numerically proved through a series of benchmark tests.