Explicit lower bounds for Stokes eigenvalue problems by using nonconforming finite elements

Abstract

An algorithm is proposed to give explicit lower bounds of the Stokes eigenvalues by utilizing two nonconforming finite element methods: Crouzeix–Raviart (CR) element and enriched Crouzeix–Raviart (ECR) element. Compared with the existing literatures which give lower eigenvalue bounds under the asymptotic condition that the mesh size is “small enough”, the proposed algorithm in this paper drops the asymptotic condition and provide explicit lower bounds even for a rough mesh. Numerical experiments are also performed to validate the theoretical results.

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Correspondence to Xuefeng Liu.

Additional information

This work is supported in part by National Science Foundations of China (NSFC 91730302, 11771434, 91330202, 11371026), Science Challenge Project (no. TZ2016002), the National Center for Mathematics and Interdisciplinary Science, CAS; X. Liu is supported by Japan Society for the Promotion of Science, Grand-in-Aid for Young Scientist (B) 26800090, Grant-in-Aid for Scientific Research (B) 16H03950.

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Xie, M., Xie, H. & Liu, X. Explicit lower bounds for Stokes eigenvalue problems by using nonconforming finite elements. Japan J. Indust. Appl. Math. 35, 335–354 (2018). https://doi.org/10.1007/s13160-017-0291-7

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Keywords

  • Stokes eigenvalue problem
  • Eigenvalue bound
  • Crouzeix–Raviart element
  • Enriched Crouzeix–Raviart element
  • Explicit lower bound

Mathematics Subject Classification

  • 65N30
  • 65N25
  • 65L15