DIV-CURL vector quasi-interpolation on a finite domain

Abstract

This paper presents a quasi-interpolation method for DIV-CURL vector splines in two dimensions on both infinite and finite domains. The quasi-interpolant is a linear combination of translates of dilates of a basis function. In particular, our discussion focuses on the approximation of a vector-valued function defined on a finite domain for practical application purposes. In such a case, edge functions are introduced for preserving the convergence of the quasi-interpolant on the boundaries. These edge functions can be determined by means of the polynomial reproduction properties of the quasi-interpolation. The analysis of convergence has shown that the quasi-interpolant defined on a regular grid of whole R2 can reproduce linear polynomial and has an O(h2 | log h|) error bound, while the modified quasi-interpolant defined on a square I2 has an O(h) error bound if the edge functions are designed for reproducing a constant.