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Verified norm estimation for the inverse of linear elliptic operators using eigenvalue evaluation

Abstract

This paper proposes a verified numerical method of proving the invertibility of linear elliptic operators. This method also provides a verified norm estimation for the inverse operators. This type of estimation is important for verified computations of solutions to elliptic boundary value problems. The proposed method uses a generalized eigenvalue problem to derive the norm estimation. This method has several advantages. Namely, it can be applied to two types of boundary conditions: the Dirichlet type and the Neumann type. It also provides a way of numerically evaluating lower and upper bounds of target eigenvalues. Numerical examples are presented to show that the proposed method provides effective estimations in most cases.

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Acknowledgments

The authors express their sincere thanks to Prof. M. Plum and Prof. K. Nagatou-Plum in Karlsruhe Institute of Technology, Germany for his valuable comment and kind remarks. They also express their appreciation for reviewer’s attentive review and valuable comments. The second author was supported by a Grant-in-Aid for JSPS Fellows. The third author was partially supported by a Grant-in-Aid for Young Scientists (B) (No. 23740092) from Japan Society for the Promotion of Science.

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Correspondence to Kazuaki Tanaka.

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Tanaka, K., Takayasu, A., Liu, X. et al. Verified norm estimation for the inverse of linear elliptic operators using eigenvalue evaluation. Japan J. Indust. Appl. Math. 31, 665–679 (2014). https://doi.org/10.1007/s13160-014-0156-2

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  • DOI: https://doi.org/10.1007/s13160-014-0156-2

Keywords

  • Eigenvalue problem
  • Elliptic operator
  • Finite element method
  • Inverse norm estimation
  • Numerical verification

Mathematics Subject Classification

  • 65N25
  • 65N30
  • 35J25