Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Uniform error estimates for certain narrow Lagrange finite elements

Author: N. Al Shenk
Journal: Math. Comp. 63 (1994), 105-119
MSC: Primary 65N30
MathSciNet review: 1226816
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Abstract: Error estimates of Dupont and Scott are used to derive uniform error estimates for Lagrange finite elements in $ {\Re ^n}\;(n \geq 2)$ under the following conditions: (1) The elements can be arbitrarily narrow in any coordinate direction such that a sufficient number of interpolation points are grouped on lines parallel to that coordinate axis, and (2) the space of approximating functions $ {F_T}$ in each element T must include the space of polynomials of degree $ \leq m - 1$ for some $ m \geq 1 + n/2$. If n is odd, this does not cover elements of lowest degree that are normally considered with the shape regularity requirement that the ratio of their outer and inner diameters be bounded. For example, if $ n = 3$, the usual requirement with shape regularity is that each $ {F_T}$ contain all first-degree polynomials. The result of this paper requires that each $ {F_T}$ contain all quadratic polynomials, and consequently does not apply to linear (Courant) elements in tetrahedrons or trilinear (tensor) elements in rectangular boxes. Counterexamples in these two cases are included.

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  • [1] I. Babuška and A. K. Aziz, On the angle condition in the finite element method, SIAM J. Numer. Anal. 13 (1976), no. 2, 214–226. MR 0455462
  • [2] Pierre Jamet, Estimations d’erreur pour des éléments finis droits presque dégénérés, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. \jname RAIRO Analyse Numérique 10 (1976), no. R-1, 43–60 (French, with Loose English summary). MR 0455282
  • [3] Robert E. Barnhill and John A. Gregory, Interpolation remainder theory from Taylor expansions on triangles, Numer. Math. 25 (1975/76), no. 4, 401–408. MR 0448948
  • [4] John A. Gregory, Error bounds for linear interpolation on triangles, The mathematics of finite elements and applications, II (Proc. Second Brunel Univ. Conf. Inst. Math. Appl., Uxbridge, 1975) Academic Press, London, 1976, pp. 163–170. MR 0458795
  • [5] Martin H. Schultz, Spline analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. Prentice-Hall Series in Automatic Computation. MR 0362832
  • [6] Lyubomir T. Dechevski and Ewald Quak, On the Bramble-Hilbert lemma, Numer. Funct. Anal. Optim. 11 (1990), no. 5-6, 485–495. MR 1079287, 10.1080/01630569008816384
  • [7] Todd Dupont and Ridgway Scott, Constructive polynomial approximation in Sobolev spaces, Recent advances in numerical analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1978) Publ. Math. Res. Center Univ. Wisconsin, vol. 41, Academic Press, New York-London, 1978, pp. 31–44. MR 519055
  • [8] N. A. Shenk, An elementary derivation of finite element error estimates, submitted to SIAM Rev.
  • [9] M. Abramowitz and I. A. Stegun, eds., Handbook of mathematical functions, Dover, New York, 1964.

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Article copyright: © Copyright 1994 American Mathematical Society