Error estimates for a vectorial second-order elliptic eigenproblem by the local L2 projected C0 finite element method

Huo-Yuan Duan, Ping Lin, Roger C. E. Tan

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    Abstract

    In this paper, a theoretical framework with a completely new theory is presented for the general curlcurl-graddiv second-order elliptic eigenproblem when discretized by the recently developed local L-2 projected C-0 finite element method. The theoretical framework consists of two Fortin-type interpolations and an Inf-Sup inequality associated with a trilinear curl/div form. From this framework, error estimates are readily derived for the source problem as well as the eigenproblem. The new theory is verified for the local L-2 projected C-0 finite element method with elementwise linear element L-2 projections applied to div and curl operators and the C-0 linear elements enriched with some face-bubbles and element-bubbles for the singular solution. The challenging question is whether C-0 elements can give spectrally correct approximations when eigenfunctions are singular (not in (H-1(Omega))(d) space). It is shown that optimal error bounds can be obtained from this theoretical framework with O(h(2r)) for eigenvalues while O(h(r)) for singular eigenfunctions in (H-r(Omega))(d) (d = 2, 3), where r <1, and that the local L-2 projected C-0 finite element method is spectrally correct.

    Original languageEnglish
    Pages (from-to)1678-1714
    Number of pages37
    JournalSIAM Journal on Numerical Analysis
    Volume51
    Issue number3
    DOIs
    Publication statusPublished - 2013

    Keywords

    • SPECTRAL APPROXIMATION
    • ELECTROMAGNETIC EIGENPROBLEMS
    • L-2 projected C-0 finite element method
    • FLUID-STRUCTURE INTERACTION
    • Fortin-type interpolation
    • Inf-Sup inequality
    • singular solution
    • EIGENVALUE PROBLEMS
    • DISCONTINUOUS GALERKIN APPROXIMATION
    • SINGULAR FIELD METHOD
    • DISCRETE COMPACTNESS
    • curlcurl-graddiv eigenproblem
    • error estimates
    • STRUCTURE SYSTEMS
    • MAXWELL EQUATIONS
    • spectral correct approximation
    • POLYHEDRAL DOMAINS

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