A priori error estimates for Lagrange interpolation on triangles

Kenta Kobayashi; Takuya Tsuchiya

Applications of Mathematics (2015)

  • Volume: 60, Issue: 5, page 485-499
  • ISSN: 0862-7940

Abstract

top
We present the error analysis of Lagrange interpolation on triangles. A new a priori error estimate is derived in which the bound is expressed in terms of the diameter and circumradius of a triangle. No geometric conditions on triangles are imposed in order to get this type of error estimates. To derive the new error estimate, we make use of the two key observations. The first is that squeezing a right isosceles triangle perpendicularly does not reduce the approximation property of Lagrange interpolation. An arbitrary triangle is obtained from a squeezed right triangle by a linear transformation. The second key observation is that the ratio of the singular values of the linear transformation is bounded by the circumradius of the target triangle.

How to cite

top

Kobayashi, Kenta, and Tsuchiya, Takuya. "A priori error estimates for Lagrange interpolation on triangles." Applications of Mathematics 60.5 (2015): 485-499. <http://eudml.org/doc/271569>.

@article{Kobayashi2015,
abstract = {We present the error analysis of Lagrange interpolation on triangles. A new a priori error estimate is derived in which the bound is expressed in terms of the diameter and circumradius of a triangle. No geometric conditions on triangles are imposed in order to get this type of error estimates. To derive the new error estimate, we make use of the two key observations. The first is that squeezing a right isosceles triangle perpendicularly does not reduce the approximation property of Lagrange interpolation. An arbitrary triangle is obtained from a squeezed right triangle by a linear transformation. The second key observation is that the ratio of the singular values of the linear transformation is bounded by the circumradius of the target triangle.},
author = {Kobayashi, Kenta, Tsuchiya, Takuya},
journal = {Applications of Mathematics},
keywords = {finite element method; Lagrange interpolation; circumradius condition; minimum angle condition; maximum angle condition; finite element method; Lagrange interpolation; circumradius condition; minimum angle condition; maximum angle condition},
language = {eng},
number = {5},
pages = {485-499},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A priori error estimates for Lagrange interpolation on triangles},
url = {http://eudml.org/doc/271569},
volume = {60},
year = {2015},
}

TY - JOUR
AU - Kobayashi, Kenta
AU - Tsuchiya, Takuya
TI - A priori error estimates for Lagrange interpolation on triangles
JO - Applications of Mathematics
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 5
SP - 485
EP - 499
AB - We present the error analysis of Lagrange interpolation on triangles. A new a priori error estimate is derived in which the bound is expressed in terms of the diameter and circumradius of a triangle. No geometric conditions on triangles are imposed in order to get this type of error estimates. To derive the new error estimate, we make use of the two key observations. The first is that squeezing a right isosceles triangle perpendicularly does not reduce the approximation property of Lagrange interpolation. An arbitrary triangle is obtained from a squeezed right triangle by a linear transformation. The second key observation is that the ratio of the singular values of the linear transformation is bounded by the circumradius of the target triangle.
LA - eng
KW - finite element method; Lagrange interpolation; circumradius condition; minimum angle condition; maximum angle condition; finite element method; Lagrange interpolation; circumradius condition; minimum angle condition; maximum angle condition
UR - http://eudml.org/doc/271569
ER -

References

top
  1. Adams, R. A., Fournier, J. J. F., Sobolev Spaces, Pure and Applied Mathematics 140 Academic Press, New York (2003). (2003) Zbl1098.46001MR2424078
  2. Babuška, I., Aziz, A. K., 10.1137/0713021, SIAM J. Numer. Anal. 13 (1976), 214-226. (1976) Zbl0324.65046MR0455462DOI10.1137/0713021
  3. Brandts, J., Korotov, S., Kříek, M., 10.1016/j.camwa.2007.11.010, Comput. Math. Appl. 55 (2008), 2227-2233. (2008) MR2413688DOI10.1016/j.camwa.2007.11.010
  4. Brenner, S. C., Scott, L. R., 10.1007/978-0-387-75934-0_7, Texts in Applied Mathematics 15 Springer, New York (2008). (2008) Zbl1135.65042MR2373954DOI10.1007/978-0-387-75934-0_7
  5. Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext Springer, New York (2011). (2011) Zbl1220.46002MR2759829
  6. Ciarlet, P. G., The Finite Element Method for Elliptic Problems, Classics in Applied Mathematics 40 SIAM, Philadelphia (2002), Repr., unabridged republ. of the orig. 1978. (2002) MR1930132
  7. Ern, A., Guermond, J.-L., 10.1007/978-1-4757-4355-5, Applied Mathematical Sciences 159 Springer, New York (2004). (2004) Zbl1059.65103MR2050138DOI10.1007/978-1-4757-4355-5
  8. Hannukainen, A., Korotov, S., Kříek, M., 10.1007/s00211-011-0403-2, Numer. Math. 120 (2012), 79-88. (2012) MR2885598DOI10.1007/s00211-011-0403-2
  9. Horn, R. A., Johnson, C. R., Topics in Matrix Analysis, Cambridge University Press, Cambridge (1991). (1991) Zbl0729.15001MR1091716
  10. Jamet, P., Estimations d'erreur pour des éléments finis droits presque dégénérés, Rev. Franc. Automat. Inform. Rech. Operat., R 10 French (1976), 43-60. (1976) MR0455282
  11. Kobayashi, K., Tsuchiya, T., 10.1007/s13160-013-0128-y, Japan J. Ind. Appl. Math. 31 (2014), 193-210. (2014) Zbl1295.65011MR3167084DOI10.1007/s13160-013-0128-y
  12. Kobayashi, K., Tsuchiya, T., 10.1007/s13160-014-0161-5, Japan J. Ind. Appl. Math. 32 (2015), 65-76. (2015) MR3318902DOI10.1007/s13160-014-0161-5
  13. Kobayashi, K., Tsuchiya, T., An extension of Babuška-Aziz's theorem to higher order Lagrange interpolation, ArXiv:1508.00119 (2015). 
  14. Kříek, M., On semiregular families of triangulations and linear interpolation, Appl. Math., Praha 36 (1991), 223-232. (1991) MR1109126
  15. Liu, X., Kikuchi, F., Analysis and estimation of error constants for P 0 and P 1 interpolations over triangular finite elements, J. Math. Sci., Tokyo 17 (2010), 27-78. (2010) Zbl1248.65118MR2676659
  16. Shenk, N. A., 10.1090/S0025-5718-1994-1226816-5, Math. Comput. 63 (1994), 105-119. (1994) Zbl0807.65003MR1226816DOI10.1090/S0025-5718-1994-1226816-5
  17. Yamamoto, T., Elements of Matrix Analysis, Japanese Saiensu-sha (2010). (2010) 
  18. Ženíšek, A., The convergence of the finite element method for boundary value problems of the system of elliptic equations, Apl. Mat. 14 Czech (1969), 355-376. (1969) Zbl0188.22604MR0245978
  19. Zlámal, M., 10.1007/BF02161362, Numer. Math. 12 (1968), 394-409. (1968) MR0243753DOI10.1007/BF02161362

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.