High order finite element calculations for the deterministic Cahn-Hilliard equation

TL;DR

This paper introduces a high-order finite element method for solving the deterministic Cahn-Hilliard equation, demonstrating efficiency and exploring effects of polynomial free energy approximation and bifurcations.

Contribution

It presents a novel high-degree continuous nodal element approach for the Cahn-Hilliard equation, outperforming traditional methods in efficiency.

Findings

Efficient high-order finite element method developed

Polynomial approximation impacts bifurcation behavior

Method compares favorably with existing strategies

Abstract

In this work, we propose a numerical method based on high degree continuous nodal elements for the Cahn-Hilliard evolution. The use of the p-version of the finite element method proves to be very efficient and favorably compares with other existing strategies (C^1 elements, adaptive mesh refinement, multigrid resolution, etc). Beyond the classical benchmarks, a numerical study has been carried out to investigate the influence of a polynomial approximation of the logarithmic free energy and the bifurcations near the first eigenvalue of the Laplace operator.