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Analysis of a Streamline-Diffusion Finite Element Method on Bakhvalov-Shishkin Mesh for Singularly Perturbed Problem

Part of: Partial differential equations, boundary value problems

Published online by Cambridge University Press:  20 February 2017

Yunhui Yin ,
Peng Zhu  and
Bin Wang
Yunhui Yin*
Affiliation:
College of Mathematics Physics and Information Engineering, Jiaxing University, Jiaxing, Zhejiang 314001, China
Peng Zhu*
Affiliation:
College of Mathematics Physics and Information Engineering, Jiaxing University, Jiaxing, Zhejiang 314001, China
Bin Wang*
Affiliation:
College of Mechanical and Electrical Engineering, Jiaxing University, Jiaxing, Zhejiang 314001, China
*
*Corresponding author. Email addresses:yunhui.yin@163.com (Y. Yin), zhupeng.hnu@gmail.com (P. Zhu), wangbin.70s@aliyun.com (B. Wang).
*Corresponding author. Email addresses:yunhui.yin@163.com (Y. Yin), zhupeng.hnu@gmail.com (P. Zhu), wangbin.70s@aliyun.com (B. Wang).
*Corresponding author. Email addresses:yunhui.yin@163.com (Y. Yin), zhupeng.hnu@gmail.com (P. Zhu), wangbin.70s@aliyun.com (B. Wang).
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Abstract

In this paper, a bilinear Streamline-Diffusion finite element method on Bakhvalov-Shishkin mesh for singularly perturbed convection – diffusion problem is analyzed. The method is shown to be convergent uniformly in the perturbation parameter ∈ provided only that ∈ ≤ N–1. An convergent rate in a discrete streamline-diffusion norm is established under certain regularity assumptions. Finally, through numerical experiments, we verified the theoretical results.

Keywords

singularly perturbed problemStreamline-Diffusion finite element methodBakhvalov-Shishkin mesherror estimate

MSC classification

Secondary: 65N15: Error bounds 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 1 , February 2017 , pp. 44 - 64
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Miller, J., O’Riordan, E., and Shishkin, G., Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore, 2000.Google Scholar
[2] Roos, H.G., Layer adapted grids for singular perturbation problems, Z. Angew. Math. Mech., vol. 78 (1998), pp. 291–309.3.0.CO;2-R>CrossRefGoogle Scholar
[3] Roos, H.G., Stynes, M., and Tobiska, L., Robust Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion-Reaction and Flow Problems, Springer, Berlin, 2008.Google Scholar
[4] Linss, T. and Stynes, M., Asymptotic analysis and Shishkin-type decomposition for an ellipyic convection-diffusion problem, J. Math. Anal. Appl., vol. 261 (2001), pp. 604–632.CrossRefGoogle Scholar
[5] Zhang, Z.M., Finite element superconvergent on shishkin mesh for 2-D convection-Diffusion problems, Mathematics of computation, vol. 72 (2003), pp. 1147–1177.CrossRefGoogle Scholar
[6] Apel, T. and Dobroeolski, M., Anisotropic interpolation with application to the finite element method, Computing, vol. 47 (1992), pp. 277–293.CrossRefGoogle Scholar
[7] Lin, Q. and Yan, N., The Construction Analysis of High Efficiency Finite Element, Hebei University Press, China, 1996 (in chinese).Google Scholar
[8] Linss, T., An upwind difference scheme on a novel Shishkin-type mesh for a linear convection-diffusion problem, J. Comput. Appl. Math., vol. 110 (1999), pp. 93–104.CrossRefGoogle Scholar
[9] Linss, T., Analysis of a Galerkin finite element on a Bakhvalov-Shishkin for a linear conmection-diffusion problem, IMA J. Numer. Anal., vol. 20 (2000), pp. 621–632.CrossRefGoogle Scholar
[10] 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., vol. 41 (2003), pp. 1620–1642.CrossRefGoogle Scholar
[11] Johnson, C. and Saranen, J., Streamline diffusion methods for the incompressible Euler and Navier-Stocks equations, Math. Comp., vol. 47 (1986), pp. 1–18.CrossRefGoogle Scholar
[12] Linss, T. and Stynes, M., The SDFEM on Shishkin meshes for linear convection-diffusion problems, Numer. Math., vol. 87 (2001), pp. 457–484.CrossRefGoogle Scholar
[13] Stynes, M. and Tobiska, L., Analysis of the streamline-diffusion finite element method on a Shishkin mesh for a convection-diffusion problem with exponential layers, J. Numer. Math., vol. 9 (2001), pp. 59–76.CrossRefGoogle Scholar
[14] Ciarlet, P. G., The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.Google Scholar
[15] Linss, T. and Stynes, M., Numerical methods on Shishkin meshes for linear convection-diffusion problems, Comput. Methods Appl. Mech. Eng., vol. 190 (2001), pp. 3527–3542.CrossRefGoogle Scholar
[16] Hughes, T.J.R. and Brooks, A., A multidimensional upwise scheme with no crossind difflusion, in Finite element methods for convection dominated flows, ASME, AMD-34 (1979), pp. 19-35.Google Scholar