Approximate Hermite quasi-interpolation

TL;DR

This paper develops approximate Hermite quasi-interpolants using function and derivative values on a grid, resulting in simple, high-order methods for solving elliptic PDEs with constant coefficients.

Contribution

It introduces new approximate Hermite quasi-interpolants that leverage function and derivative data for high-order approximation of elliptic PDE solutions.

Findings

High-order accuracy in approximating solutions to elliptic equations

Simple formulas for practical implementation

Effective use of function and derivative data at grid points

Abstract

In this paper we derive approximate quasi-interpolants when the values of a function $u$ and of some of its derivatives are prescribed at the points of a uniform grid. As a byproduct of these formulas we obtain very simple approximants which provide high order approximations for solutions to elliptic differential equations with constant coefficients.