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Inverse-type estimates on $ hp$-finite element spaces and applications


Author: Emmanuil H. Georgoulis
Journal: Math. Comp. 77 (2008), 201-219
MSC (2000): Primary 65J05; Secondary 65R20
DOI: https://doi.org/10.1090/S0025-5718-07-02068-6
Published electronically: September 18, 2007
MathSciNet review: 2353949
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Abstract: This work is concerned with the development of inverse-type inequalities for piecewise polynomial functions and, in particular, functions belonging to $ hp$-finite element spaces. The cases of positive and negative Sobolev norms are considered for both continuous and discontinuous finite element functions.The inequalities are explicit both in the local polynomial degree and the local mesh size.The assumptions on the $ hp$-finite element spaces are very weak, allowing anisotropic (shape-irregular) elements and varying polynomial degree across elements. Finally, the new inverse-type inequalities are used to derive bounds for the condition number of symmetric stiffness matrices of $ hp$-boundary element method discretisations of integral equations, with element-wise discontinuous basis functions constructed via scaled tensor products of Legendre polynomials.


References [Enhancements On Off] (What's this?)

  • 1. J. BERGH AND J. LÖFSTRÖM, Interpolation spaces. An introduction, Springer-Verlag, Berlin, 1976.
    Grundlehren der Mathematischen Wissenschaften, No. 223. MR 0482275 (58:2349)
  • 2. V. I. BURENKOV, Sobolev spaces on Domains, 2000.
    Teubner.
  • 3. P. G. CIARLET,
    The finite element method for elliptic problems, vol. 40 of Classics in Applied Mathematics.
    Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002.
    Reprint of the 1978 original. MR 1930132
  • 4. W. DAHMEN, B. FAERMANN, I. GRAHAM, W. HACKBUSCH, AND S. SAUTER, Inverse inequalities on non-quasi-uniform meshes and application to the mortar element method, Math. Comp. 73 (2004), pp. 1107-1138 MR 2047080 (2005d:65187)
  • 5. I. G. GRAHAM, W. HACKBUSCH, AND S. A. SAUTER, Finite elements on degenerate meshes: Inverse-type inequalities and applications, IMA Journal of Numerical Analysis 25 (2005), 379-407. MR 2126208 (2006b:65183)
  • 6. I. G. GRAHAM AND W. MCLEAN, Anisotropic mesh refinement, the conditioning of Galerkin boundary element matrices and simple preconditioners, SIAM Journal of Numerical Analysis 44 (2006), 1487-1513. MR 2257114
  • 7. W. MCLEAN, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000. MR 1742312 (2001a:35051)
  • 8. C. SCHWAB, $ p$- and $ hp$-finite element methods, Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 1998.
    Theory and applications in solid and fluid mechanics. MR 1695813 (2000d:65003)

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Additional Information

Emmanuil H. Georgoulis
Affiliation: Department of Mathematics, University of Leicester, University Road, Leicester, LE1 7RH, United Kingdom
Email: Emmanuil.Georgoulis@mcs.le.ac.uk

DOI: https://doi.org/10.1090/S0025-5718-07-02068-6
Keywords: Inverse-type inequalities, $hp$-finite element spaces, condition number, boundary element methods
Received by editor(s): February 27, 2006
Received by editor(s) in revised form: December 4, 2006
Published electronically: September 18, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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