Extending Babuška-Aziz’s theorem to higher-order Lagrange interpolation
Tóm tắt
We consider the error analysis of Lagrange interpolation on triangles and tetrahedrons. For Lagrange interpolation of order one, Babuška and Aziz showed that squeezing a right isosceles triangle perpendicularly does not deteriorate the optimal approximation order. We extend their technique and result to higher-order Lagrange interpolation on both triangles and tetrahedrons. To this end, we make use of difference quotients of functions with two or three variables. Then, the error estimates on squeezed triangles and tetrahedrons are proved by a method that is a straightforward extension of the original one given by Babuška-Aziz.
Tài liệu tham khảo
R. A. Adams, J. J. F. Fournier: Sobolev Spaces. Pure and Applied Mathematics 140, Academic Press, New York, 2003.
T. Apel: Anisotropic Finite Elements: Local Estimates and Applications. Advances in Numerical Mathematics, Teubner, Stuttgart, 1999.
K. E. Atkinson: An Introduction to Numerical Analysis. John Wiley & Sons, New York, 1989.
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.
S. C. Brenner, L. R. Scott: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics 15, Springer, New York, 2008.
H. Brezis: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext, Springer, New York, 2011.
P. G. Ciarlet: The Finite Element Method for Elliptic Problems. Repr., unabridged republ. of the 1978 orig. Classics in Applied Mathematics 40, SIAM, Philadelphia, 2002.
R. G. Durán: Error estimates for 3-d narrow finite elements. Math. Comput. 68 (1999), 187–199.
A. Ern, J.-L. Guermond: Theory and Practice of Finite Elements. AppliedMathematical Sciences 159, Springer, New York, 2004.
P. Jamet: Estimations d’erreur pour des éléments finis droits presque dégénérés. Rev. Franc. Automat. Inform. Rech. Operat. 10, Analyse numer., R-1 10 (1976), 43–60. (In French.)
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. Křížek: On semiregular families of triangulations and linear interpolation. Appl. Math., Praha 36 (1991), 223–232.
M. Křížek: On the maximum angle condition for linear tetrahedral elements. SIAM J. Numer. Anal. 29 (1992), 513–520.
A. Kufner, O. John, S. Fučík: Function Spaces. Monographs and Textsbooks on Mechanics of Solids and Fluids, Noordhoff International Publishing, Leyden; Publishing House of the Czechoslovak Academy of Sciences, Prague, 1977.
O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Ural’tseva: Linear and Quasilinear Equations of Parabolic Type. Translated from Russian original. Translations of Mathematical Monographs 23, AMS, Providence, 1968.
N. A. Shenk: Uniform error estimates for certain narrow Lagrange finite elements. Math. Comput. 63 (1994), 105–119.
T. Yamamoto: Introduction to Numerical Analysis. Saiensu-sha, 2003. (In Japanese.)