Bivariate Shepard-Bernoulli operators

TL;DR

This paper introduces an extension of Shepard-Bernoulli operators to bivariate data, utilizing local basis functions and generalized Taylor polynomials, improving approximation without special data structures.

Contribution

The paper develops a new bivariate Shepard-Bernoulli operator using local basis functions and generalized Taylor polynomials, avoiding complex data partitioning.

Findings

Comparable accuracy to QSHEP2D and CSHEP2D schemes

No need for special partitions or structured data

Deep analysis of approximation properties

Abstract

In this paper we extend the Shepard-Bernoulli operators introduced in [6] to the bivariate case. These new interpolation operators are realized by using local support basis functions introduced in [23] instead of classical Shepard basis functions and the bivariate three point extension [13] of the generalized Taylor polynomial introduced by F. Costabile in [11]. The new operators do not require either the use of special partitions of the node convex hull or special structured data as in [8]. We deeply study their approximation properties and provide an application to the scattered data interpolation problem; the numerical results show that this new approach is comparable with the other well known bivariate schemes QSHEP2D and CSHEP2D by Renka [34, 35].