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.

翻译：在对矿物分离进程的模型应用中,我们建议对卡赫-希利亚德方程式进行数字分析,采用自动可变稳定限量元素(AVS-FE)的时时分分法。AVS-FE法是一种Petrov-Galerkin法,采用Demkowicz和Gopalakrishnan的不连续性Petrov-Galerkin法的最佳不连续测试功能概念。但是,试验空间由H1和H(div)等全球连续的Hilbert空间组成。因此,AVS-FE近似采用经典C0或Raviart-ThoomasFE法基础功能。最佳测试功能保证了AVS-FE法的数值稳定性,并导致具有对称性和肯定性的离散性系统。因此,AVS-FE法方法可以在没有限制CFL条件的情况下在空间时段尺寸的情况下解决卡赫-Hilard方程式的等式方程式。我们使用传统的H2和Ravart-TFA的精确度标准标准标准标准,也显示LS的精确度的校正结果。