Numerische Mathematik

, Volume 82, Issue 3, pp 351–388

The approximation theory for the p-version finite element method and application to non-linear elliptic PDEs

  • Mark Ainsworth
  • David Kay

Abstract.

Approximation theoretic results are obtained for approximation using continuous piecewise polynomials of degree p on meshes of triangular and quadrilateral elements. Estimates for the rate of convergence in Sobolev spaces \(W^{m,q}(\Omega)\), \(q\in [1,\infty]\) are given. The results are applied to estimate the rate of convergence when the p-version finite element method is used to approximate the \(\alpha\)-Laplacian. It is shown that the rate of convergence of the p-version is always at least that of the h-version (measured in terms of number of degrees of freedom used). If the solution is very smooth then the p-version attains an exponential rate of convergence. If the solution has certain types of singularity, the rate of convergence of the p-version is twice that of the h-version. The analysis generalises the work of Babuska and others to the case \(q\not=2\). In addition, the approximation theoretic results find immediate application for some types of spectral and spectral element methods.

Mathematics Subject Classification (1991):65N15, 65N30 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Mark Ainsworth
    • 1
  • David Kay
    • 1
  1. 1. Mathematics Department, Leicester University, Leicester LE1 7RH, UK; e-mail: ain@mcs.le.ac.uk GB

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