Basic error estimates for elliptic problems

This is a survey of techniques for the interpolation of scattered data in three or more independent variables. It covers schemes that can be used for any number of variables as well as schemes specifically designed for three variables. Emphasis is on breadth rather than depth, but there are explicit illustrations of different techniques used in the solution of multivariate interpolation problems.

Scattered data interpolation in three or more variables

For A a triangulated d-dimensional region in Rd, let Sr (A) denote the vector space of all cr functions F on A that, restricted to any simplex in A, are given by polynomials of degree at most m. We consider the problem of computing the dimension of such spaces. We develop a homological approach to this problem and apply it specifically to the case of triangulated manifolds A in the plane, getting lower bounds on the dimension of Sr (A) for all r. For r = 1, we prove a conjecture of Strang concerning the generic dimension of the space of Cl splines over a triangulated manifold in R2. Finally, we consider the space of continuous piecewise linear functions over nonsimplicial decompositions of a plane region.

/pdf/homology-of-smooth-splines-generic-triangulations-and-a-362cosgqzb.pdf

Homology of smooth splines: generic triangulations and a conjecture of Strang

We consider spaces of piecewise polynomials of degreed defined over a triangulation of a polygonal domain and possessingr continuous derivatives globally. Morgan and Scott constructed a basis in the case wherer=1 andd≥5. The purpose of this paper is to extend the dimension part of their result tor≥0 andd≥4r+l. We use Bezier nets as a crucial tool in deriving the dimension of such spaces.

The dimension of bivariate spline spaces of smoothnessr for degreed≥4r+1

The purpose of this paper is to discuss the dimension of linear spaces of piecewise polynomials defined on triangulations of the plane. Such spaces are of considerable interest in general approximation and data fitting, as well as in the numerical solution of boundary-value problems by the finite-element method. We begin by defining the notion of triangulation.

On the Dimension of Spaces Of Piecewise Polynomials in Two Variables

/pdf/a-nodal-basis-for-piecewise-polynomials-of-degree-5-5aclysiu41.pdf

A nodal basis for ¹ piecewise polynomials of degree ≥5

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A nodal basis for ¹ piecewise polynomials of degree ≥5