A framework of verified eigenvalue bounds for self-adjoint differential operators

Xuefeng Liu

Research output: Contribution to journalArticle

20 Citations (Scopus)

Abstract

For eigenvalue problems of self-adjoint differential operators, a universal framework is proposed to give explicit lower and upper bounds for the eigenvalues. In the case of the Laplacian operator, by applying Crouzeix-Raviart finite elements, an efficient algorithm is developed to bound the eigenvalues for the Laplacian defined in 1D, 2D and 3D spaces. Moreover, for nonconvex domains, for which case there may exist singularities of eigenfunctions around re-entrant corners, the proposed algorithm can easily provide eigenvalue bounds. By further adopting the interval arithmetic, the explicit eigenvalue bounds from numerical computations can be mathematically correct.

Original languageEnglish
JournalApplied Mathematics and Computation
DOIs
Publication statusAccepted/In press - 2015

Fingerprint

Eigenvalue Bounds
Self-adjoint Operator
Differential operator
Eigenvalue
Interval Arithmetic
Explicit Bounds
Eigenvalues and eigenfunctions
Numerical Computation
Eigenvalue Problem
Eigenfunctions
Upper and Lower Bounds
Efficient Algorithms
Singularity
Finite Element
Framework

Keywords

  • Eigenvalue bounds
  • Non-conforming finite element method
  • Quantitative error estimation
  • Self-adjoint differential operator
  • Verified computation

ASJC Scopus subject areas

  • Applied Mathematics
  • Computational Mathematics

Cite this

A framework of verified eigenvalue bounds for self-adjoint differential operators. / Liu, Xuefeng.

In: Applied Mathematics and Computation, 2015.

Research output: Contribution to journalArticle

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