Abstract
Recently, the so-called circumradius condition (or estimate) was derived, which is a new estimate of the W 1,p-error of linear Lagrange interpolation on triangles in terms of their circumradius. The published proofs of the estimate are rather technical and do not allow clear, simple insight into the results. In this paper, we give a simple direct proof of the p = ∞ case. This allows us to make several observations such as on the optimality of the circumradius estimate. Furthermore, we show how the case of general p is in fact nothing more than a simple scaling of the standard O(h) estimate under the maximum angle condition, even for higher order interpolation. This allows a direct interpretation of the circumradius estimate and condition in the context of the well established theory of the maximum angle condition.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
I. Babuška, A. K. Aziz: On the angle condition in the finite element method. SIAM J. Numer. Anal. 13 (1976), 214–226.
R. E. Barnhill, J. A. Gregory: Sard kernel theorems on triangular domains with application to finite element error bounds. Numer. Math. 25 (1976), 215–229.
P.G. Ciarlet: The finite element method for elliptic problems. Studies in Mathematics and Its Applications. Vol. 4, North-Holland Publishing Company, Amsterdam, 1978.
P. J. Davis: Interpolation and Approximation. Dover Books on Advanced Mathematics, Dover Publications, New York, 1975.
A. Hannukainen, S. Korotov, M. Krížek: The maximum angle condition is not necessary for convergence of the finite element method. Numer. Math. 120 (2012), 79–88.
P. Jamet: Estimations d’erreur pour des éléments finis droits presque dégénérés. Rev. Franc. Automat. Inform. Rech. Operat. 10, Analyse numer., R-1, (1976), 43–60. (In French.)
K. Kobayashi: On the interpolation constants over triangular elements. RIMS Kokyuroku 1733 (2011), 58–77. (In Japanese.)
K. Kobayashi, T. Tsuchiya: A Babuška-Aziz type proof of the circumradius condition. Japan. J. Ind. Appl. Math. 31 (2014), 193–210.
K. Kobayashi, T. Tsuchiya: A priori error estimates for Lagrange interpolation on triangles. Appl. Math., Praha 60 (2015), 485–499.
M. Krížek: On semiregular families of triangulations and linear interpolation. Appl. Math., Praha 36 (1991), 223–232.
V. Kucera: On necessary and sufficient conditions for finite element convergence. Submitted to Numer. Math. http://arxiv.org/abs/1601.02942 (preprint).
A. Rand: Delaunay refinement algorithms for numerical methods. Ph.D. thesis, www.math.cmu.edu/~arand/papers/arand thesis.pdf, Carnegie Mellon University, 2009.
A. Ženíšek: The convergence of the finite element method for boundary value problems of the system of elliptic equations. Apl. Mat. 14 (1969), 355–376. (In Czech.) zbl
M. Zlámal: On the finite element method. Numer. Math. 12 (1968), 394–409.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work is a part of the research project P201/13/00522S of the Czech Science Foundation. V. Kučera is currently a Fulbright visiting scholar at Brown University, Providence, RI, USA, supported by the J.William Fulbright Commission in the Czech Republic.
Rights and permissions
About this article
Cite this article
Kučera, V. Several notes on the circumradius condition. Appl Math 61, 287–298 (2016). https://doi.org/10.1007/s10492-016-0132-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10492-016-0132-z