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

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.