Convergence analysis of Galerkin finite element approximations to shape gradients in eigenvalue optimization

被引：11
|
作者：
Zhu, Shengfeng [1 ,2 ]
Hu, Xianliang [3 ]
Liao, Qifeng [4 ]
机构：
[1] East China Normal Univ, Sch Math Sci, Dept Data Math, Shanghai 200241, Peoples R China
[2] East China Normal Univ, Sch Math Sci, Shanghai Key Lab Pure Math & Math Practice, Shanghai 200241, Peoples R China
[3] Zhejiang Univ, Sch Math Sci, Hangzhou 310027, Zhejiang, Peoples R China
[4] ShanghaiTech Univ, Sch Informat Sci & Technol, Shanghai 201210, Peoples R China
来源：
BIT NUMERICAL MATHEMATICS | 2020年 / 60卷 / 03期
基金：
中国国家自然科学基金; 上海市自然科学基金;
关键词：
Shape optimization; Shape gradient; Eigenvalue problem; Finite element; Error estimate; Multiple eigenvalue; LEVEL SET METHODS; DESIGN SENSITIVITY;
D O I：
10.1007/s10543-019-00782-3
中图分类号：
TP31 [计算机软件];
学科分类号：
081202 ; 0835 ;
摘要：
This paper concerns the accuracy of Galerkin finite element approximations to two types of shape gradients for eigenvalue optimization. Under certain regularity assumptions on domains, a priori error estimates are obtained for the two approximate shape gradients. Our convergence analysis shows that the volume integral formula converges faster and offers higher accuracy than the boundary integral formula. Numerical experiments validate the theoretical results for the problem with a pure Dirichlet boundary condition. For the problem with a pure Neumann boundary condition, the boundary formulation numerically converges as fast as the distributed type.
引用
收藏
页码：853 / 878
页数：26
相关论文
共 50 条
  • [1] Convergence analysis of Galerkin finite element approximations to shape gradients in eigenvalue optimization
    Shengfeng Zhu
    Xianliang Hu
    Qifeng Liao
    BIT Numerical Mathematics, 2020, 60 : 853 - 878
  • [2] Convergence analysis of mixed finite element approximations to shape gradients in the Stokes equation
    Zhu, Shengfeng
    Gao, Zhiming
    COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING, 2019, 343 : 127 - 150
  • [3] Convergence analysis for eigenvalue approximations on triangular finite element meshes
    Todorov, TD
    NUMERICAL ANALYSIS AND ITS APPLICATIONS, 2005, 3401 : 558 - 565
  • [4] On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems
    Caorsi, S
    Fernandes, P
    Raffetto, M
    SIAM JOURNAL ON NUMERICAL ANALYSIS, 2000, 38 (02) : 580 - 607
  • [5] On mixed finite element approximations of shape gradients in shape optimization with the Navier-Stokes equation
    Li, Jiajie
    Zhu, Shengfeng
    Shen, Xiaoqin
    NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, 2023, 39 (02) : 1604 - 1634
  • [6] A priori and a posteriori error analysis for discontinuous Galerkin finite element approximations of biharmonic eigenvalue problems
    Wang, Liang
    Xiong, Chunguang
    Wu, Huibin
    Luo, Fusheng
    ADVANCES IN COMPUTATIONAL MATHEMATICS, 2019, 45 (5-6) : 2623 - 2646
  • [7] A priori and a posteriori error analysis for discontinuous Galerkin finite element approximations of biharmonic eigenvalue problems
    Liang Wang
    Chunguang Xiong
    Huibin Wu
    Fusheng Luo
    Advances in Computational Mathematics, 2019, 45 : 2623 - 2646
  • [8] SHAPE OPTIMIZATION FOR STOKES FLOWS: A FINITE ELEMENT CONVERGENCE ANALYSIS
    Fumagalli, Ivan
    Parolini, Nicola
    Verani, Marco
    ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE, 2015, 49 (04): : 921 - 951
  • [9] Postprocessing and higher order convergence for the mixed finite element approximations of the eigenvalue problem
    Chen, Hongtao
    Jia, Shanghui
    Xie, Hehu
    APPLIED NUMERICAL MATHEMATICS, 2011, 61 (04) : 615 - 629
  • [10] L-INFINITY-CONVERGENCE OF FINITE-ELEMENT GALERKIN APPROXIMATIONS FOR PARABOLIC PROBLEMS
    NITSCHE, JA
    RAIRO-ANALYSE NUMERIQUE-NUMERICAL ANALYSIS, 1979, 13 (01): : 31 - 54
12345