# Article

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Keywords:
finite element method; a priori error estimate; circumradius condition; Lagrange interpolation
Summary:
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 $p=\infty$ 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.
References:
[1] Babuška, I., Aziz, A. K.: On the angle condition in the finite element method. SIAM J. Numer. Anal. 13 (1976), 214-226. DOI 10.1137/0713021 | MR 0455462 | Zbl 0324.65046
[2] Barnhill, R. E., Gregory, J. A.: Sard kernel theorems on triangular domains with application to finite element error bounds. Numer. Math. 25 (1976), 215-229. DOI 10.1007/BF01399411 | MR 0458000 | Zbl 0304.65076
[3] Ciarlet, P. G.: The finite element method for elliptic problems. Studies in Mathematics and Its Applications. Vol. 4 North-Holland Publishing Company, Amsterdam (1978). MR 0520174 | Zbl 0383.65058
[4] Davis, P. J.: Interpolation and Approximation. Dover Books on Advanced Mathematics Dover Publications, New York (1975). MR 0380189 | Zbl 0329.41010
[5] Hannukainen, A., Korotov, S., Křížek, M.: The maximum angle condition is not necessary for convergence of the finite element method. Numer. Math. 120 (2012), 79-88. DOI 10.1007/s00211-011-0403-2 | MR 2885598 | Zbl 1255.65196
[6] Jamet, P.: Estimations d'erreur pour des éléments finis droits presque dégénérés. Rev. Franc. Automat. Inform. Rech. Operat. {\it 10}, Analyse numer., R-1, (1976), 43-60 French. MR 0455282
[7] Kobayashi, K.: On the interpolation constants over triangular elements. RIMS Kokyuroku Japanese 1733 (2011), 58-77.
[8] Kobayashi, K., Tsuchiya, T.: A Babuška-Aziz type proof of the circumradius condition. Japan. J. Ind. Appl. Math. 31 (2014), 193-210. DOI 10.1007/s13160-013-0128-y | MR 3167084 | Zbl 1295.65011
[9] Kobayashi, K., Tsuchiya, T.: A priori error estimates for Lagrange interpolation on triangles. Appl. Math., Praha 60 (2015), 485-499. DOI 10.1007/s10492-015-0108-4 | MR 3396477 | Zbl 1363.65015
[10] K{ř}{í}{ž}ek, M.: On semiregular families of triangulations and linear interpolation. Appl. Math., Praha 36 (1991), 223-232. MR 1109126 | Zbl 0728.41003
[11] Kučera, V.: On necessary and sufficient conditions for finite element convergence. Submitted to Numer. Math. http://arxiv.org/abs/1601.02942 (preprint).
[12] Rand, A.: Delaunay refinement algorithms for numerical methods. Ph.D. thesis, www.math.cmu.edu/ {arand/papers/arand_thesis.pdf} Carnegie Mellon University (2009). MR 2713254
[13] Ženíšek, A.: The convergence of the finite element method for boundary value problems of the system of elliptic equations. Apl. Mat. 14 (1969), 355-376 Czech. MR 0245978 | Zbl 0188.22604
[14] Zl{á}mal, M.: On the finite element method. Numer. Math. 12 (1968), 394-409. DOI 10.1007/BF02161362 | MR 0243753 | Zbl 0176.16001

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