High-Precision Eigenvalue Bound for the Laplacian with Singularities

Xuefeng Liu (), Tomoaki Okayama and Shin’ichi Oishi
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Xuefeng Liu: Waseda University, Research Institute for Science and Engineering
Tomoaki Okayama: Hitotsubashi University, Graduate School of Economics
Shin’ichi Oishi: Waseda University, Faculty of Science and Engineering

A chapter in Computer Mathematics, 2014, pp 311-323 from Springer

Abstract: Abstract For the purpose of bounding eigenvalues of the Laplacian over a bounded polygonal domain, we propose an algorithm to give high-precision bound even in the case that the eigenfunction has singularities around reentrant corners. The algorithm is a combination of the finite element method and the Lehmann–Goerisch theorem. The interval arithmetic is adopted in floating point number computation. Since all the error in the computation, e.g., the function approximation error, the floating point number rounding error, are exactly estimated, the result can be mathematically correct. In the end of the chapter, there are computational examples over an L-shaped domain and a square-minus-square domain that demonstrate the efficiency of our proposed algorithm.

Keywords: Eigenvalue problem; Elliptic operator; Finite element method; Verified computation; Lehmann-Goerisch’s theorem (search for similar items in EconPapers)
Date: 2014
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-662-43799-5_23

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DOI: 10.1007/978-3-662-43799-5_23

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