We prove anisotropic interpolation error estimates for quadrilateral and hexahedral elements with all possible shape function spaces, which cover the intermediate families, tensor product families and serendipity families. Moreover, we show that the anisotropic interpolation error estimates hold for derivatives of any order. This goal is accomplished by investigating an interpolation defined via orthogonal expansions.
@article{bwmeta1.element.doi-10_2478_s11533-013-0203-2,
author = {Mingxia Li and Shipeng Mao},
title = {Anisotropic interpolation error estimates via orthogonal expansions},
journal = {Open Mathematics},
volume = {11},
year = {2013},
pages = {621-629},
zbl = {1269.65016},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0203-2}
}
Mingxia Li; Shipeng Mao. Anisotropic interpolation error estimates via orthogonal expansions. Open Mathematics, Tome 11 (2013) pp. 621-629. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0203-2/
[1] Acosta G., Apel T., Durán R.G., Lombardi A.L., Anisotropic error estimates for an interpolant defined via moments, Computing, 2008, 82(1), 1–9 http://dx.doi.org/10.1007/s00607-008-0259-1 | Zbl 1154.65007
[2] Apel T., Anisotropic Finite Elements: Local Estimates and Applications, Advances in Numerical Mathematics, Teubner, Stuttgart, 1999 | Zbl 0917.65090
[3] Apel T., Dobrowolski M., Anisotropic interpolation with applications to the finite element method, Computing, 1992, 47(3–4), 277–293 http://dx.doi.org/10.1007/BF02320197 | Zbl 0746.65077
[4] Brenner S.C., Scott L.R., The Mathematical Theory of Finite Element Methods, Texts Appl. Math., 15, Springer, New York, 1994 | Zbl 0804.65101
[5] Chen S., Shi D., Zhao Y., Anisotropic interpolation and quasi-Wilson element for narrow quadrilateral meshes, IMA J. Numer. Anal., 2004, 24(1), 77–95 http://dx.doi.org/10.1093/imanum/24.1.77 | Zbl 1049.65129
[6] Chen S., Zheng Y., Mao S., Anisotropic error bounds of Lagrange interpolation with any order in two and three dimensions, Appl. Math. Comput., 2011, 217(22), 9313–9321 http://dx.doi.org/10.1016/j.amc.2011.04.015 | Zbl 1222.41004
[7] Ciarlet P.G., The Finite Element Method for Elliptic Problems, Stud. Math. Appl., 4, North-Holland, Amsterdam-New York-Oxford, 1978 | Zbl 0383.65058
[8] Girault V., Raviart P.-A., Finite Element Methods for Navier-Stokes equations, Springer Ser. Comput. Math., 5, Springer, Berlin, 1986 | Zbl 0413.65081
[9] Hannukainen A., Korotov S., Křížek M., The maximum angle condition is not necessary for convergence of the finite element method, Numer. Math., 2012, 120(1), 79–88 http://dx.doi.org/10.1007/s00211-011-0403-2 | Zbl 1255.65196
[10] Křížek M., On semiregular families of triangulations and linear interpolation, Appl. Math., 1991, 36(3), 223–232 | Zbl 0728.41003
[11] Luke Y.L., The Special Functions and their Approximations I, Math. Sci. Eng., 53, Academic Press, New York-London, 1969 | Zbl 0193.01701
[12] Mao S., Shi Z., Error estimates of triangular finite elements under a weak angle condition, J. Comput. Appl. Math., 2009, 230(1), 329–331 http://dx.doi.org/10.1016/j.cam.2008.11.008 | Zbl 1168.65063
[13] Sansone G., Orthogonal Functions, Dover, New York, 1991