Linear interpolation on a triangle
This page is dedicated to estimates for the error in
linear interpolation
(interpolation by linear polynomials) to (function values at) the vertices of
a triangle (and more generally a simplex). Great amounts of effort have been
lavished on this problem by numerical analysts because of the connection with
finite elements (linear interpolation at the vertices of a triangle is
Courant's finite element), and by approximation theorists because it is the
simplest nontrivial example of multivariate Lagrange interpolation
(see here).
Some recent work and people involved
 David Handscomb (dch@comlab.oxford.ac.uk):
using the variational calculus to get sharp L_2bounds on the error in
linear interpolation on a triangle
 Thomas Sauer (sauer@helena.mi.unierlangen.de):
computational aspects of multivariate Lagrange interpolation and
associated errors (with Yuan Xu)
 Pavel Shvartsman (pshv@math.technion.ac.il):
provides a
nice proof
 Yuri Subbotin (yunsub@imm.eburg.su):
dependence of estimates of a
multidimensional piecewise polynomial approximation on the geometric
characteristics of the triangulation
 Shayne Waldron
(waldron@math.auckland.ac.nz):
integral error formula for linear interpolation on a triangle and the
corresponding L_perror bounds
 Yuan Xu (yuan@bright.uoregon.edu):
computational aspects of multivariate Lagrange interpolation and the
associated errors (with Sauer)
References
 Some relevant books
 Linear approximation, A. Sard, AMS monograph (1963)
 The finite element method for elliptic problems, P. G. Ciarlet (1978)
 Some older papers which are still of interest
 Error bounds for linear interpolation on a triangle, J. A. Gregory,
In: Mathematics of finite elements and applications (J. Whiteman, ed.)
(1975)
 General Lagrange and Hermite interpolation in R^n with applications
to finite element methods, P. G. Ciarlet and P. A. Raviart, Arch. Rational
Mech. Anal. 46 (1972), pp 177199
 Sur l'evaluation de l'erreur d'interpolation de Lagrange dans un
ouvert de R^n, R. Arcangeli and J. L. Gout, Rev. Francaise Automat.
Informat. Rech. Oper., Anal. Numer. 10(3) (1976), pp 527

Other sources of information
This document is maintained by
Shayne
(waldron@math.auckland.ac.nz).
Last Modified: .