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finite element method; Lagrange interpolation; circumradius condition; minimum angle condition; maximum angle condition
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.
[1] Adams, R. A., Fournier, J. J. F.: Sobolev Spaces. Pure and Applied Mathematics 140 Academic Press, New York (2003). MR 2424078 | Zbl 1098.46001
[2] 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
[3] Brandts, J., Korotov, S., Kříek, M.: On the equivalence of regularity criteria for triangular and tetrahedral finite element partitions. Comput. Math. Appl. 55 (2008), 2227-2233. DOI 10.1016/j.camwa.2007.11.010 | MR 2413688
[4] Brenner, S. C., Scott, L. R.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics 15 Springer, New York (2008). DOI 10.1007/978-0-387-75934-0_7 | MR 2373954 | Zbl 1135.65042
[5] Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext Springer, New York (2011). MR 2759829 | Zbl 1220.46002
[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. MR 1930132
[7] Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Applied Mathematical Sciences 159 Springer, New York (2004). DOI 10.1007/978-1-4757-4355-5 | MR 2050138 | Zbl 1059.65103
[8] Hannukainen, A., Korotov, S., Kříek, 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
[9] Horn, R. A., Johnson, C. R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991). MR 1091716 | Zbl 0729.15001
[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. MR 0455282
[11] 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
[12] Kobayashi, K., Tsuchiya, T.: On the circumradius condition for piecewise linear triangular elements. Japan J. Ind. Appl. Math. 32 (2015), 65-76. DOI 10.1007/s13160-014-0161-5 | MR 3318902
[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. MR 1109126
[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. MR 2676659 | Zbl 1248.65118
[16] Shenk, N. A.: Uniform error estimates for certain narrow Lagrange finite elements. Math. Comput. 63 (1994), 105-119. DOI 10.1090/S0025-5718-1994-1226816-5 | MR 1226816 | Zbl 0807.65003
[17] Yamamoto, T.: Elements of Matrix Analysis. Japanese Saiensu-sha (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. MR 0245978 | Zbl 0188.22604
[19] Zlámal, M.: On the finite element method. Numer. Math. 12 (1968), 394-409. DOI 10.1007/BF02161362 | MR 0243753
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