论文标题
Cahn-Hilliard方程的时空解决方案的稳定FE方法
A Stable FE Method For the Space-Time Solution of the Cahn-Hilliard Equation
论文作者
论文摘要
在应用于矿物分离过程的建模时，我们通过采用自动变异稳定有限元（AVS-FE）方法的时空离散来提出CAHN-HILLIARD方程的数值分析。 AVS-FE方法是一种Petrov-Galerkin方法，它采用了Demkowicz和Gopalakrishnan的不连续性彼得 - 加勒金（DPG）方法的最佳不连续测试函数的概念。但是，试验空间由H1和H（DIV）等全球连续的希尔伯特空间组成。因此，AVS-FE近似采用经典的C0或Raviart-Thomas Fe基函数。最佳测试函数保证了AVS-FE方法的数值稳定性，并导致对称和正定确定的离散系统。因此，AVS-FE方法可以在空间和时间上求解Cahn-Hilliard方程，而无需限制的CFL条件来决定时空元素的大小。我们介绍了空间中一个和二维问题的数值验证。验证显示L2和H1规范中收敛的最佳速率。还提供了使用AVS-FE方法的内置误差估计器的网格自适应改进结果。
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.
