Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

Sharp anisotropic interpolation error estimates for rectangular Raviart-Thomas elements


Author: Martin Stynes
Journal: Math. Comp. 83 (2014), 2675-2689
MSC (2010): Primary 65N30; Secondary 65N12, 65N15
DOI: https://doi.org/10.1090/S0025-5718-2014-02826-3
Published electronically: March 24, 2014
MathSciNet review: 3246805
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Anisotropic $ L_p$-norm error estimates are derived for the standard rectangular Raviart-Thomas elements $ RT_{[k]}(\tilde K)$ in $ \mathbb{R}^d$ for $ p\in [1, \infty ],\ k\ge 0$ and $ d \ge 2$. Here $ \tilde K$ is an affine image of an axi-parallel parallelotope $ K$. The proofs are based on a variant of the classical Poincaré inequality. The estimates derived make full use of the asymmetric nature of the vector space components of $ RT_{[k]}(\tilde K)$; a Shishkin mesh example demonstrates their superiority over previous estimates.


References [Enhancements On Off] (What's this?)

  • [1] Gabriel Acosta, Thomas Apel, Ricardo G. Durán, and Ariel L. Lombardi, Error estimates for Raviart-Thomas interpolation of any order on anisotropic tetrahedra, Math. Comp. 80 (2011), no. 273, 141-163. MR 2728975 (2011m:65262), https://doi.org/10.1090/S0025-5718-2010-02406-8
  • [2] Gabriel Acosta and Ricardo G. Durán, The maximum angle condition for mixed and nonconforming elements: application to the Stokes equations, SIAM J. Numer. Anal. 37 (1999), no. 1, 18-36 (electronic). MR 1721268 (2000g:65107), https://doi.org/10.1137/S0036142997331293
  • [3] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957 (56 #9247)
  • [4] Douglas N. Arnold, Daniele Boffi, and Richard S. Falk, Quadrilateral $ H({\rm div})$ finite elements, SIAM J. Numer. Anal. 42 (2005), no. 6, 2429-2451 (electronic). MR 2139400 (2006d:65129), https://doi.org/10.1137/S0036142903431924
  • [5] Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205 (92d:65187)
  • [6] C. Clavero, J. L. Gracia, and E. O'Riordan, A parameter robust numerical method for a two dimensional reaction-diffusion problem, Math. Comp. 74 (2005), no. 252, 1743-1758. MR 2164094 (2006e:65192), https://doi.org/10.1090/S0025-5718-05-01762-X
  • [7] Daniele Boffi, Franco Brezzi, Leszek F. Demkowicz, Ricardo G. Durán, Richard S. Falk, and Michel Fortin, Mixed finite elements, compatibility conditions, and applications, Lecture Notes in Mathematics, vol. 1939, Springer-Verlag, Berlin, 2008. Lectures given at the C.I.M.E. Summer School held in Cetraro, June 26-July 1, 2006; Edited by Boffi and Lucia Gastaldi. MR 2459075 (2010h:65219)
  • [8] Ricardo G. Durán and Ariel L. Lombardi, Error estimates for the Raviart-Thomas interpolation under the maximum angle condition, SIAM J. Numer. Anal. 46 (2008), no. 3, 1442-1453. MR 2391001 (2009b:65302), https://doi.org/10.1137/060665312
  • [9] R. Hiptmair, Canonical construction of finite elements, Math. Comp. 68 (1999), no. 228, 1325-1346. MR 1665954 (2000b:65214), https://doi.org/10.1090/S0025-5718-99-01166-7
  • [10] Runchang Lin and Martin Stynes, A balanced finite element method for singularly perturbed reaction-diffusion problems, SIAM J. Numer. Anal. 50 (2012), no. 5, 2729-2743. MR 3022240, https://doi.org/10.1137/110837784
  • [11] Fang Liu, Niall Madden, Martin Stynes, and Aihui Zhou, A two-scale sparse grid method for a singularly perturbed reaction-diffusion problem in two dimensions, IMA J. Numer. Anal. 29 (2009), no. 4, 986-1007. MR 2557053 (2010m:65282), https://doi.org/10.1093/imanum/drn048
  • [12] J.M. Thomas, Sur l'analyse numerique des methodes d'elements finis hybrides et mixtes, Ph.D. thesis, Université Pierre et Marie Curie, Paris, 1977.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65N30, 65N12, 65N15

Retrieve articles in all journals with MSC (2010): 65N30, 65N12, 65N15


Additional Information

Martin Stynes
Affiliation: Department of Mathematics, National University of Ireland, Cork, Ireland
Address at time of publication: Applied Mathematics Division, Beijing Computational Science Research Center, No. 3 He-Qing Road, Hai-Dian District, Beijing 100084, China
Email: m.stynes@ucc.ie

DOI: https://doi.org/10.1090/S0025-5718-2014-02826-3
Received by editor(s): August 12, 2012
Received by editor(s) in revised form: April 16, 2013
Published electronically: March 24, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society