Exact asymptotics of the optimal $L_{p,\Omega}$-error of linear spline interpolation

TL;DR

This paper derives the exact asymptotic behavior of the best weighted Lp-error for linear spline interpolation of smooth functions with positive Hessian, and provides an optimal triangulation algorithm.

Contribution

It offers the precise asymptotics of the optimal weighted Lp-error and an algorithm for constructing asymptotically optimal triangulations.

Findings

Exact asymptotics of the optimal error for linear spline interpolation.

Algorithm for constructing asymptotically optimal triangulations.

Computed the minimum Lp-error for quadratic functions over unit-area triangles.

Abstract

In this paper we provide the exact asymptotics of the optimal weighted $L_p$-error, $0<p< \infty$, of linear spline interpolation of $C^2$ functions with positive Hessian. The full description of the behavior of the optimal error leads to the algorithm for construction of an asymptotically optimal sequence of triangulations. In addition, we compute the minimum of the $L_p$-error of linear interpolation of the function $x^2+y^2$ over all triangles of unit area for all $0<p<\infty$. This provides the exact constant in the asymptotics of the optimal error.