The Approximation Theory for thep-Version of the Finite Element Method

Abstract

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 domains 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_.

The $p$-version of the finite element method for elliptic equations of order $2l$

“…(1.6) This improves upon the rate of convergence found in [16] (and [9]- [10]) which is optimal up to an arbitrarily small e > 0. In Section 2, we describe the notation used and our model problem.…”

“…As mentioned in the introduction, the results correspondîng to Theorem 4.1 in [9], [16] are based on the assumption that u e O, which is not the usual resuit predicted by elliptic regularity theory. In the previous section, we analyzed the approximation of fonctions which were known to be in H k (ft) n HQ(CI), k>2l --.…”

“…By L 2 (ft) = H°(n) and H k (il) 9 k> 0 we dénote the standard Sobolev spaces (with index 2). Also, HQ(£1) dénotes the subspace of functions with k -1 vanishing normal derivatives on F. For k >• 0 note an integer, we define H k (£l), HQ(CI) as the usual interpolation spaces (by the A'-method, see [7] Hp ER (R (K ) ) <= H k (R (K) ) will dénote the space of all periodic fonctions with period 2 K. By ^p(n), respectively TSJCR(K)), we dénote the space of all algebraic, respectively trigonométrie (with period 2 K) , polynomials of total degree at most p on £1, respectively R(K).…”

“…In proving (1.5), we do not use the interpolation assumption used in [9], [16]. The use of this assumption, however, allows us to extend (1.5) to the case f<=fc^2f-i.…”

The $h-p$ version of the finite element method with quasiuniform meshes

“…The first theoretical paper appeared in 1981 (see [6]). See also [2,5,7,10,11,14] for most recent results. For the numerical, computational, implementational and engineering aspects of the h-p version we refer to [3,[21][22][23][24],…”