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.
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Received August 2, 1995 / Revised version received January 26, 1998
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Ainsworth, M., Kay, D. The approximation theory for the p-version finite element method and application to non-linear elliptic PDEs. Numer. Math. 82, 351–388 (1999). https://doi.org/10.1007/s002110050423
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DOI: https://doi.org/10.1007/s002110050423