THE APPROXIMATION-THEORY FOR THE P-VERSION OF THE FINITE-ELEMENT METHOD

The purpose of this article is to present the approximation theory underlying the p-version of the finite element method. By exploiting the relationship between polynomial approximation and certain weighted Sobolev spaces, which are identified as the domain of positive real powers of the Legendre operator, this one-dimensional result is generalized via a tensor product construction to yield a nonconforming piecewise polynomial approximation result in the usual unweighted Sobolev spaces on triangulated domains of R'. It is then shown that essentially the same result holds for approximation by conforming piecewise polynomials provided that the function being approximated possesses the same degree of conformality across the common boundaries of adjacent simplices and the same homogeneous boundary conditions. Inverse results are given for the special case of approximation in L//2.