In its application to the modeling of a mineral separation process, we propose the numerical analysis of the Cahn-Hilliard equation by employing space-time 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 $H^1(\mathrm{\Omega })$ and $H(\text{div},\mathrm{\Omega })$. Hence, the AVS-FE approximations employ classical $C^0$ 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 multiple numerical verifications of both stationary and transient problems. The verifications show optimal rates of convergence in $L^2(\mathrm{\Omega })$ and $H^1(\mathrm{\Omega })$ norms. Results for mesh adaptive refinements in both space and time using a built-in error estimator of the AVS-FE method are also presented.

在将其应用于矿物分离过程的建模中，我们通过使用自动变分稳定有限元（AVS-FE）方法的时空离散化，提出了Cahn-Hilliard方程的数值分析。AVS-FE方法是Petrov-Galerkin方法，它采用了Demkowicz和Gopalakrishnan提出的间断Petrov-Galerkin（DPG）方法的最佳间断测试功能的概念。但是，试验空间由全局连续的希尔伯特空间组成，例如$H^{\mathrm{1个}}（\mathrm{\Omega }）$ 和 $H（\text{股利}，\mathrm{\Omega }）$。因此，AVS-FE近似采用经典$C^0$或Raviart-Thomas有限元基础函数。最佳测试函数可确保AVS-FE方法的数值稳定性，并导致离散的系统对称且为正定。因此，AVS-FE方法可以在时空上求解Cahn-Hilliard方程，而没有用于约束时空元素大小的限制性CFL条件。我们提出了稳态和瞬态问题的多个数值验证。验证显示最佳收敛速度为${\mathrm{大号}}^{\mathrm{2个}}（\mathrm{\Omega }）$ 和 $H^{\mathrm{1个}}（\mathrm{\Omega }）$规范。还介绍了使用AVS-FE方法的内置误差估计器在空间和时间上进行网格自适应细化的结果。