A Stable FE Method For the Space-Time Solution of the Cahn-Hilliard Equation
A Stable FE Method For the Space-Time Solution of the Cahn-Hilliard Equation
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DOI：
10.1016/j.jcp.2021.110426
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发表时间：
2020-06
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影响因子：
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通讯作者：
Eirik Valseth;A. Romkes;Austin R. Kaul
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文献类型：
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作者：
Eirik Valseth;A. Romkes;Austin R. Kaul
在其应用到矿物分离过程的建模中，我们提出了Cahn-Hilliard方程的数值分析，采用时空离散的自动变分稳定有限元（AVS-FE）方法。AVS-FE方法是一种Petrov-Galerkin方法，它采用Demkowicz和Gopalakrishnan的间断Petrov-Galerkin（DPG）方法的最佳间断测试函数的概念。然而，试探空间由全局连续的希尔伯特空间组成，如H 1（Ω）和H（div，Ω）。因此，AVS-FE近似采用经典的C 0或Raviart-Thomas FE基函数。最佳的测试功能保证AVS-FE方法的数值稳定性，并导致离散系统是对称和正定的。因此，AVS-FE方法可以在空间和时间上求解Cahn-Hilliard方程，而不需要限制CFL条件来决定时空单元的大小。我们提出了多个固定和瞬态问题的数值验证。证明了在L2（Ω）和H1（Ω）范数下的最优收敛速度.网格自适应细化在空间和时间使用内置的误差估计的AVS-FE方法的结果。
In its application to the modeling of a mineral separation process, we propose the numerical analysis of the Cahn-Hilliard equation by employing 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 (Ω) and H (div, Ω). 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 (Ω) and H 1 (Ω) 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.