The dimension of piecewise polynomial spaces, and one-sided approximation

Abstract

Two separate problems are discussed. One is a question implicit in the whole theory of piecewise polynomials: suppose we consider the space S of all piecewise polynomials of degree p and of continuity class C^q, say on a given triangulation in the plane. Then what is the dimension of S, and what is a convenient basis for this space? The answer is known in a dozen special cases, but not in general. The second question has arisen in the approximation of variational inequalities, but is of independent interest. We are given a nonnegative function u on a domain Ω, and want to approximate it from below by a nonnegative spline or finite element u_h: o ≤ u_h ≤ u. We sketch a proof that under this constraint the usual order of approximation is still possible.

This research was supported by the National Science Foundation (P22928).