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

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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.

Keywords

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

Mathematics Subject Classification

65N30 65N25 65L15 

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Copyright information

© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Center for Applied MathematicsTianjin UniversityTianjinChina
  2. 2.LSEC, ICMSEC, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  3. 3.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingChina
  4. 4.Graduate School of Science and TechnologyNiigata UniversityNiigataJapan

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