Verified eigenvalue evaluation for the laplacian over polygonal domains of arbitrary shape

Xuefeng Liu, Shinichi Oishi

    Research output: Contribution to journalArticle

    28 Citations (Scopus)

    Abstract

    The finite element method (FEM) is applied to bound leading eigenvalues of the Laplace operator over polygonal domains. Compared with classical numerical methods, most of which can only give concrete eigenvalue bounds over special domains of symmetry, our proposed algorithm can provide concrete eigenvalue bounds for domains of arbitrary shape, even when the eigenfunction has a singularity. The problem of eigenvalue estimation is solved in two steps. First, we construct a computable a priori error estimation for the FEM solution of Poisson's problem, which holds even for nonconvex domains with reentrant corners. Second, new computable lower bounds are developed for the eigenvalues. Because the interval arithmetic is implemented throughout the computation, the desired eigenvalue bounds are expected to be mathematically correct. We illustrate several computation examples, such as the cases of an L-shaped domain and a crack domain, to demonstrate the efficiency and flexibility of the proposed method.

    Original languageEnglish
    Pages (from-to)1634-1654
    Number of pages21
    JournalSIAM Journal on Numerical Analysis
    Volume51
    Issue number3
    DOIs
    Publication statusPublished - 2013

    Fingerprint

    Concretes
    Eigenvalue Bounds
    Eigenvalue
    Finite element method
    Evaluation
    Arbitrary
    Eigenvalues and eigenfunctions
    Error analysis
    Mathematical operators
    Numerical methods
    Cracks
    Finite Element Method
    Poisson Problem
    Interval Arithmetic
    Laplace Operator
    Error Estimation
    Eigenfunctions
    Crack
    Flexibility
    Numerical Methods

    Keywords

    • Eigenvalue problem
    • Elliptic operator
    • Finite element method
    • Min-max principle
    • Prager-Synge's theorem
    • Verified computation

    ASJC Scopus subject areas

    • Numerical Analysis

    Cite this

    Verified eigenvalue evaluation for the laplacian over polygonal domains of arbitrary shape. / Liu, Xuefeng; Oishi, Shinichi.

    In: SIAM Journal on Numerical Analysis, Vol. 51, No. 3, 2013, p. 1634-1654.

    Research output: Contribution to journalArticle

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