Exact asymptotics of the uniform error of interpolation by multilinear splines

TL;DR

This paper derives the exact asymptotic behavior of the uniform approximation error when interpolating C2 functions with multilinear splines, providing insights for adaptive mesh generation in various computational fields.

Contribution

It establishes the precise asymptotics of the optimal uniform error for multilinear spline interpolation of C2 functions, advancing understanding of approximation accuracy.

Findings

Derived exact asymptotics for uniform interpolation error

Analyzed optimal error behavior for multilinear splines

Applicable to adaptive mesh generation in computational geometry

Abstract

The question of adaptive mesh generation for approximation by splines has been studied for a number of years by various authors. The results have numerous applications in computational and discrete geometry, computer aided geometric design, finite element methods for numerical solutions of partial differential equations, image processing, and mesh generation for computer graphics, among others. In this paper we will investigate the questions regarding adaptive approximation of C2 functions with arbitrary but fixed throughout the domain signature by multilinear splines. In particular, we will study the asymptotic behavior of the optimal error of the weighted uniform approximation by interpolating and quasi-interpolating multilinear splines.