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求解带有非线性边界条件的涡流方程的A-φ解耦有限元格式

王然1, 张怀1, 康彤2   

  1. 1. UCAS, 中国科学院大学, 计算地球动力学重点实验室, 北京 100049;
    2. CUC, 中国传媒大学, 数据科学与智能媒体学院, 北京 100024
  • 收稿日期:2019-03-27 出版日期:2021-02-15 发布日期:2021-02-04
  • 基金资助:
    国家重点研发计划资助(编号2020YFA0713401),国家自然科学基金(编号42074108,41904067,41725017)和中央高校基本科研业务费专项资金资助.

王然, 张怀, 康彤. 求解带有非线性边界条件的涡流方程的A-φ解耦有限元格式[J]. 计算数学, 2021, 43(1): 33-55.

Wang Ran, Zhang Huai, Kang Tong. A-φ DECOUPLED FINITE ELEMENT SCHEME FOR EDDY CURRENT EQUATIONS WITH A NONLINEAR BOUNDARY CONDITION[J]. Mathematica Numerica Sinica, 2021, 43(1): 33-55.

A-φ DECOUPLED FINITE ELEMENT SCHEME FOR EDDY CURRENT EQUATIONS WITH A NONLINEAR BOUNDARY CONDITION

Wang Ran1, Zhang Huai1, Kang Tong2   

  1. 1. UCAS, Key Laboratory of Computational Geodynamics, University of Chinese Academy of Sciences, Beijing 100049, China;
    2. CUC, School of Data Science and Media Intelligence, Communication University of China, Beijing 100024, China
  • Received:2019-03-27 Online:2021-02-15 Published:2021-02-04
本文研究边界条件符合幂指数型非线性关系H × n = n × (|E × n|α-1E × n)(0 < α ≤ 1)的涡流方程.使用A-φ耦合有限元格式数值求解这类问题具有较高精度,但计算开销大. A-φ解耦有限元计算格式能够在每个时间步上分别求解矢量A和标量φ,以此降低计算规模,提高计算效率.我们证明了解耦格式中解的存在唯一性,并且给出了它的误差估计.最后给出的数值实验证明了本文所提供的解耦算法是稳定和有效的.
In this contribution, the nonlinear degenerate boundary condition of eddy current equations is obedient to a power-law nonlinearity of the form H × n = n × (|E × n|α-1E × n), α ∈ (0, 1]. Solving such problems numerically via A-φ coupled finite element scheme has higher precision, but the computation is expensive. The A-φ decoupled finite element scheme has to solve respectively two separate algebraic equation systems to the vector A and the scalar φ at each time step. The existence and uniqueness of the solution of the decoupled scheme is proved and its error estimate is discussed. Finally some numerical results are shown to verify that our scheme is feasible and efficient.

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