Galerkin and streamline diffusion finite element methods on a Shishkin mesh for a convection-diffusion problem with corner singularities
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- by Sebastian Franz, R. Bruce Kellogg and Martin Stynes PDF
- Math. Comp. 81 (2012), 661-685 Request permission
Abstract:
An error analysis of Galerkin and streamline diffusion finite element methods for the numerical solution of a singularly perturbed convection-diffusion problem is given. The problem domain is the unit square. The solution contains boundary layers and corner singularities. A tensor product Shishkin mesh is used, with piecewise bilinear trial functions. The error bounds are uniform in the singular perturbation parameter. Numerical results supporting the theory are given.References
- Thomas Apel, Anisotropic finite elements: local estimates and applications, Advances in Numerical Mathematics, B. G. Teubner, Stuttgart, 1999. MR 1716824
- P. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O’Riordan, and G. I. Shishkin, Robust computational techniques for boundary layers, Applied Mathematics (Boca Raton), vol. 16, Chapman & Hall/CRC, Boca Raton, FL, 2000. MR 1750671
- Sebastian Franz and Torsten Linß, Superconvergence analysis of the Galerkin FEM for a singularly perturbed convection-diffusion problem with characteristic layers, Numer. Methods Partial Differential Equations 24 (2008), no. 1, 144–164. MR 2371352, DOI 10.1002/num.20245
- S. Franz, T. Linß, and H.-G. Roos, Superconvergence analysis of the SDFEM for elliptic problems with characteristic layers, Appl. Numer. Math. 58 (2008), no. 12, 1818–1829. MR 2464813, DOI 10.1016/j.apnum.2007.11.005
- R. Bruce Kellogg and Martin Stynes, Corner singularities and boundary layers in a simple convection-diffusion problem, J. Differential Equations 213 (2005), no. 1, 81–120. MR 2139339, DOI 10.1016/j.jde.2005.02.011
- R. Bruce Kellogg and Martin Stynes, Sharpened bounds for corner singularities and boundary layers in a simple convection-diffusion problem, Appl. Math. Lett. 20 (2007), no. 5, 539–544. MR 2303990, DOI 10.1016/j.aml.2006.08.001
- Q. Lin, A rectangle test for finite element analysis, Proc. Syst. Sci. Eng., Great Wall (H.K.) Culture Publish Co., 1991, pp. 213–216.
- Torsten Linß, Uniform superconvergence of a Galerkin finite element method on Shishkin-type meshes, Numer. Methods Partial Differential Equations 16 (2000), no. 5, 426–440. MR 1778398, DOI 10.1002/1098-2426(200009)16:5<426::AID-NUM2>3.3.CO;2-I
- T. Linß and M. Stynes, Numerical methods on Shishkin meshes for linear convection-diffusion problems, Comput. Methods Appl. Mech. Engrg. 190 (2001), 3527–3542.
- Hans-Görg Roos, Martin Stynes, and Lutz Tobiska, Robust numerical methods for singularly perturbed differential equations, 2nd ed., Springer Series in Computational Mathematics, vol. 24, Springer-Verlag, Berlin, 2008. Convection-diffusion-reaction and flow problems. MR 2454024
- Grigory I. Shishkin and Lidia P. Shishkina, Difference methods for singular perturbation problems, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 140, CRC Press, Boca Raton, FL, 2009. MR 2454526
- Martin Stynes and Lutz Tobiska, The SDFEM for a convection-diffusion problem with a boundary layer: optimal error analysis and enhancement of accuracy, SIAM J. Numer. Anal. 41 (2003), no. 5, 1620–1642. MR 2035000, DOI 10.1137/S0036142902404728
- Zhimin Zhang, Finite element superconvergence on Shishkin mesh for 2-D convection-diffusion problems, Math. Comp. 72 (2003), no. 243, 1147–1177. MR 1972731, DOI 10.1090/S0025-5718-03-01486-8
Additional Information
- Sebastian Franz
- Affiliation: Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland
- MR Author ID: 745061
- Email: sebastian.franz@ul.ie
- R. Bruce Kellogg
- Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
- Email: bklandrum@gmail.com
- Martin Stynes
- Affiliation: Department of Mathematics, National University of Ireland, Cork, Ireland
- Email: m.stynes@ucc.ie
- Received by editor(s): September 3, 2009
- Received by editor(s) in revised form: November 2, 2010, and January 25, 2011
- Published electronically: July 20, 2011
- Additional Notes: The research of the first author was supported by Science Foundation Ireland under the Research Frontiers Programme 2008; Grant 08/RFP/MTH1536
The research of the second author was supported by the Boole Centre for Research in Informatics at National University of Ireland, Cork - © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 81 (2012), 661-685
- MSC (2010): Primary 65N30
- DOI: https://doi.org/10.1090/S0025-5718-2011-02526-3
- MathSciNet review: 2869032