The approximation theory for the p-version of the finite element method.

*(English)*Zbl 0572.65074In this paper the author presents the approximation theoretical results which are needed to prove convergence estimates for the so-called p- version of finite element theory. These are the sort of results which are proved, for example, with reference to explicit calculations on triangles by I. Babuška, B. A. Szabo and I. N. Katz [ibid. 18, 515- 545 (1981; Zbl 0487.65059)]. The author considers the Legendre ordinary differential operator \(L=-d/dt[(1-t^ 2)d/dt]\) on [-1,1] and the weighted Sobolev norms
\[
\| u\|^ 2_{Z^ k}=\int^{1}_{- 1}| u|^ 2dt+\int^{1}_{-1}| \frac{d^ ku}{dt^ k}|^ 2(1-t^ 2)^ kdt,
\]
where the norms are extended for non- integral k. Because the eigenvalues and eigenfunctions of L are well- known, and because the eigenfunctions are polynomials, several approximation theorems are available. For example, the author shows that for \(s>s'\geq 0\), \(s,s'\neq 1/2+\) an integer, and u with finite norm, for each nonnegative integer p there is a polynomial \(\phi_ p\) of degree p such that \(\| u-\phi_ p\|_{Z^{s'}}\leq Cp^{-s+s'}\| u\|_{Z^ s}.\) (This estimate is extended to more than one dimension using tensor products.) The inverse of this theorem up to an arbitrarily small \(\epsilon\) is also established. The result cited above involves polynomials which may not have continuous derivatives across element boundaries and is measured in a weighted norm. The author extends the estimates to the usual Sobolev norms and to piecewise polynomials which have continuous derivatives across element boundaries at the cost of an arbitrarily small \(\epsilon\) added to the exponent of p.

Reviewer: M.Sussman

##### MSC:

65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

34B05 | Linear boundary value problems for ordinary differential equations |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |