by Shayne Waldron 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: This article is available in: 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 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: This article is available in: 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 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: This article is available in: 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 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: This article is available in: 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 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: This article is available in: 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

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: This article is available in: 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 Last updated: 25 March 1997 Status: Appeared in the SIAM Journal on Numerical Analysis, Volume 35, Number 3 (1998), 1191-1200 Availability: This article is available in: 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 Status: Appeared in the SIAM Journal on Numerical Analysis, Volume 35, Number 3 (1998), 1191-1200 Availability: This article is available in: 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 Availability: This article is available in: 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 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 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 Here is the final SIAM version