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_2-bounds on the error in
linear interpolation on a triangle
- Thomas Sauer (sauer@helena.mi.uni-erlangen.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.e-burg.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_p-error 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 177-199
- 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 5-27
-
Other sources of information
This document is maintained by
Shayne
(waldron@math.auckland.ac.nz).
Last Modified: .