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

  • Shengfeng ZhuEmail author
  • Xianliang Hu
  • Qifeng Liao


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.


Shape optimization Shape gradient Eigenvalue problem Finite element Error estimate Multiple eigenvalue 

Mathematics Subject Classification

49Q10 65N25 65N30 



This work was supported in part by Natural Science Foundation of Shanghai (Grant No. 19ZR1414100), Science and Technology Commission of Shanghai Municipality (No. 18dz2271000) and the National Natural Science Foundation of China under Grants 11201153, 11571115 and 11601329.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Data Mathematics and Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, School of Mathematical SciencesEast China Normal UniversityShanghaiChina
  2. 2.School of Mathematical SciencesZhejiang UniversityHangzhouChina
  3. 3.School of Information Science and TechnologyShanghaiTech UniversityShanghaiChina

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