A Stable FE Method For the Space-Time Solution of the Cahn-Hilliard Equation

TL;DR

This paper introduces a stable finite element method for solving the Cahn-Hilliard equation in space-time, enabling stable, efficient simulations without restrictive time-step conditions, verified through numerical experiments.

Contribution

It develops a novel AVS-FE method employing optimal test functions that guarantees stability and symmetry, allowing for space-time solutions without CFL restrictions.

Findings

Achieves optimal convergence rates in L2 and H1 norms.

Produces symmetric, positive definite discrete systems.

Effectively handles adaptive mesh refinement.

Abstract

In its application to the modeling of a mineral separation process, we propose the numerical analysis of the Cahn-Hilliard equation by employing spacetime discretizations of the automatic variationally stable finite element (AVS-FE) method. The AVS-FE method is a Petrov-Galerkin method which employs the concept of optimal discontinuous test functions of the discontinuous Petrov-Galerkin (DPG) method by Demkowicz and Gopalakrishnan. The trial space, however, consists of globally continuous Hilbert spaces such as H1 and H(div). Hence, the AVS-FE approximations employ classical C0 or Raviart-Thomas FE basis functions. The optimal test functions guarantee the numerical stability of the AVS-FE method and lead to discrete systems that are symmetric and positive definite. Hence, the AVS-FE method can solve the Cahn-Hilliard equation in both space and time without a restrictive CFL condition to dictate the space-time element size. We present numerical verifications of both one and two dimensional problems in space. The verifications show optimal rates of convergence in L2 and H1 norms. Results for mesh adaptive refinements using a built-in error estimator of the AVS-FE method are also presented.