Hostname: page-component-76fb5796d-x4r87 Total loading time: 0 Render date: 2024-04-26T08:16:34.728Z Has data issue: false hasContentIssue false
  • English
  • Français

Superconvergence of a Galerkin FEM for Higher-Order Elements in Convection-Diffusion Problems

Published online by Cambridge University Press:  28 May 2015

Sebastian Franz  and
H.-G. Roos
Sebastian Franz*
Affiliation:
Institut für Numerische Mathematik, Technische Universität Dresden, 01062 Dresden, Germany
H.-G. Roos*
Affiliation:
Institut für Numerische Mathematik, Technische Universität Dresden, 01062 Dresden, Germany
*
Corresponding author.Email address:sebastian.franz@tu-dresden.de
Corresponding author.Email address:hans-goerg.roos@tu-dresden.de
Get access

Abstract

In this paper we present a first supercloseness analysis for higher-order Galerkin FEM applied to a singularly perturbed convection-diffusion problem. Using a solution decomposition and a special representation of our finite element space, we are able to prove a supercloseness property of p + 1/4 in the energy norm where the polynomial order p ≥ 3 is odd.

Keywords

65N1265N3065N50Singular perturbationlayer-adapted meshessuperconvergencepostprocessing
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 7 , Issue 3 , August 2014 , pp. 356 - 373
Copyright
Copyright © Global Science Press Limited 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Apel, T.. Anisotropic finite elements: local estimates and applications. Advances in Numerical Mathematics. B. G.Teubner Stuttgart, 1999.Google Scholar
[2]Axelsson, O. and Gustafsson, I.. Quasioptimal finite element approximations of first order hyperbolic and of convection-dominated convection-diffusion equations. In Frank O. Axelsson, L.S. and Van Der Sluis, A., editors, Analytical and Numerical Approaches to Asymptotic Problems in Analysis Proceedings of the Conference on Analytical and Numerical Approaches to Asymptotic Problems, volume 47 of North-Holland Mathematics Studies, pages 273–280. North-Holland, 1981.Google Scholar
[3]Franz, S.. Continuous interior penalty method on a Shishkin mesh for convection-diffusion problems with characteristic boundary layers. Comput. Meth. Appl. Mech. Engng., 197(45-48):3679–3686, 2008.CrossRefGoogle Scholar
[4]Franz, S.. Convergence Phenomena of Qp-Elements for Convection-Diffusion Problems. Numer. Methods Partial Differential Equations, 29(1):280–296, 2013.Google Scholar
[5]Franz, S.. Superconvergence using pointwise interpolation in convection-diffusion problems. Appl. Numer. Math., 76:132–144, 2014.CrossRefGoogle Scholar
[6]Franz, S. and Linß, T.. Superconvergence analysis of the Galerkin FEM for a singularly perturbed convection-diffusion problem with characteristic layers. Numer. Methods Partial Differential Equations, 24(1):144–164, 2008.CrossRefGoogle Scholar
[7]Franz, S., Linß, T., and Roos, H.-G.. Superconvergence analysis of the SDFEM for elliptic problems with characteristic layers. Appl. Numer. Math., 58(12):1818–1829, 2008.CrossRefGoogle Scholar
[8]Franz, S., Linß, T., Roos, H.-G., and Schiller, S.. Uniform superconvergence of a finite element method with edge stabilization for convection-diffusion problems. J. Comp. Math., 28(1):32–44, 2010.Google Scholar
[9]Franz, S. and Matthies, G.. Local projection stabilisation on S-type meshes for convection-diffusion problems with characteristic layers. Computing, 87(3-4):135–167, 2010.CrossRefGoogle Scholar
[10]Franz, S. and Matthies, G.. Convergence on layer-adapted meshes and anisotropic interpolation error estimates of non-standard higher order finite elements. Appl. Numer. Math., 61:723–737, 2011.CrossRefGoogle Scholar
[11]Girault, V. and Raviart, P.A.. Finite element methods for Navier-Stokes equations: theory and algorithms. Springer series in computational mathematics. Springer-Verlag, Berlin, Heidelberg, New York, 1986.Google Scholar
[12]Lin, Q., Yan, N., and Zhou, A.. A rectangle test for interpolated element analysis. In Proc. Syst. Sci. Eng., pages 217–229. Great Wall (H.K.) Culture Publish Co., 1991.Google Scholar
[13]Linß, T.. Uniform superconvergence of a Galerkin finite element method on Shishkin-type meshes. Numer. Methods Partial Differential Equations, 16(5):426–440, 2000.3.0.CO;2-R>CrossRefGoogle Scholar
[14]Ludwig, L. and Roos, H.-G.. Finite element superconvergence on Shishkin meshes for convection-diffusion problems with corner singularities. IMA J. Numer. Anal., 2013. doi:10.1093/imanum/drt027.Google Scholar
[15]Roos, H.-G. and Schopf, M.. Analysis of finite element methods on Bakhvalov-type meshes for linear convection-diffusion problems in 2d. Appl. of Mathematics, 57:97–108, 2012.CrossRefGoogle Scholar
[16]Roos, H.-G., Stynes, M., and Tobiska, L.. Robust numerical methods for singularly perturbed differential equations, volume 24 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, second edition, 2008.Google Scholar
[17]Roos, H.-G. and Zarin, H.. A supercloseness result for the discontinuous Galerkin stabilization of convection-diffusion problems on Shishkin meshes. Numer. Methods Partial Differential Equations, 23(6):1560–1576, 2007.CrossRefGoogle Scholar
[18]Stynes, M. and Tobiska, L.. The SDFEM for a convection-diffusion problem with a boundary layer: Optimal error analysis and enhancement of accuracy. SIAM J. Numer. Anal., 41(5):1620–1642, 2003.CrossRefGoogle Scholar
[19]Stynes, M. and Tobiska, L.. Using rectangular Qp elements in the SDFEM for a convection-diffusion problem with a boundary layer. Appl. Numer. Math., 58(12):1709–1802, 2008.CrossRefGoogle Scholar
[20]Yan, N.. Superconvergence analysis and a posteriori error estimation in finite element methods,volume 40 of Series in Information and Computational Science. Science Press, Beijing, 2008.Google Scholar
[21]Zarin, H.. Continuous-discontinuous finite element method for convection-diffusion problems with characteristic layers. J. Comput. Appl. Math., 231(2):626–636, 2009.CrossRefGoogle Scholar
[22]Zhang, Zh.. Finite element superconvergence approximation for one-dimensional singularly perturbed problems. Numer. Methods Partial Differential Equations, 18(3):374–395, 2002.CrossRefGoogle Scholar
[23]Zhang, Zh.. Finite element superconvergence on Shishkin mesh for 2-d convection-diffusion problems. Math. Comp., 72(423):1147–1177, 2003.CrossRefGoogle Scholar