The error in linear interpolation at the vertices of a simplex

by Shayne Waldron

Abstract:

A new formula for the error in a map which interpolates to function values at some set $\Theta\subset\Rn$ from a space of functions which contains the linear polynomials is given. From it {\it sharp} pointwise $L_\infty$-bounds for the error in linear interpolation (interpolation by linear polynomials) to (function values at) the vertices of a simplex are obtained. The corresponding `envelope theorem' giving the optimal recovery of functions is discussed.

This error formula reflects the geometry in a particularly appealing way. The error at any point $x$ not lying on a line connecting points in $\Theta$ is the sum over distinct points $v,w\in\Theta$ of $1/2$ the average of the second order derivative $D_{v-w}D_{w-v}f$ over the triangle with vertices $x,v,w$ multiplied by some function which vanishes at all of the points in $\Theta$.

Keywords: Lagrange interpolation, linear interpolation on a triangle, sharp error bounds, finite elements, Courant's finite element, multipoint Taylor formula, Kowalewski's remainder, multivariate form of Hardy's inequality, optimal recovery of functions, envelope theorems

Math Review Classification: 41A10, 41A44, 41A80 (primary), 41A05, 65N30 (secondary)

Length: 21 pages (+2 pages of details not included in the paper). There are also some open problems related to this paper, which have been submitted separately.

Comment: Written in TeX, contains 6 figures. This paper won first prize in the competition for best student papers presented at the annual meeting of the Southeastern-Atlantic Section of SIAM, Charleston, South Carolina, March 24--25, 1995. This work was partiallly supported by the Steenbock Professorship of Carl de Boor and the Center for the Mathematical Sciences at the University of Wisconsin-Madison.

Last updated: 25 March 1997

Status: Appeared in the SIAM Journal on Numerical Analysis, Volume 35, Number 3 (1998), 1191-1200

Availability:

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Here is the original long article in Postscript / pdf. The original has more details, explanation and figures, than what will appear in print.
Here is the shorter version in Postscript / pdf which is very close to what will appear. Not included (in either version) is the following section univariate examples.
Here is the final SIAM version