Tailored finite point method based on exponential bases for convection-diffusion-reaction equation

Han, Houde; Huang, Zhongyi

doi:10.1090/S0025-5718-2012-02616-0

Tailored finite point method based on exponential bases for convection-diffusion-reaction equation
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by Houde Han and Zhongyi Huang;

Math. Comp. 82 (2013), 213-226

DOI: https://doi.org/10.1090/S0025-5718-2012-02616-0

Published electronically: June 6, 2012

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Abstract:

In this paper, we propose a class of new tailored finite point methods (TFPM) for the numerical solution of a type of convection-diffusion-reaction problems in two dimensions. Our finite point method has been tailored based on the local exponential basis functions. Furthermore, our TFPM satisfies the discrete maximum principle automatically. We also study the error estimates of our TFPM. We prove that our TFPM can achieve good accuracy even when the mesh size $h\gg \varepsilon$ for some cases without any prior knowledge of the boundary layers. Our numerical examples show the efficiency and reliability of our method.

References

M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, National Bureau of Standards, 1964.
Alan E. Berger, Hou De Han, and R. Bruce Kellogg, A priori estimates and analysis of a numerical method for a turning point problem, Math. Comp. 42 (1984), no. 166, 465–492. MR 736447, DOI 10.1090/S0025-5718-1984-0736447-2
Iliya Brayanov and Ivanka Dimitrova, Uniformly convergent high-order schemes for a 2D elliptic reaction-diffusion problem with anisotropic coefficients, Numerical methods and applications, Lecture Notes in Comput. Sci., vol. 2542, Springer, Berlin, 2003, pp. 395–402. MR 2053277, DOI 10.1007/3-540-36487-0_{4}4
Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845, DOI 10.1090/gsm/019
I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 6th ed., Academic Press, Inc., San Diego, CA, 2000. Translated from the Russian; Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. MR 1773820
Houde Han and Zhongyi Huang, A tailored finite point method for the Helmholtz equation with high wave numbers in heterogeneous medium, J. Comput. Math. 26 (2008), no. 5, 728–739. MR 2444729
Houde Han and Zhongyi Huang, Tailored finite point method for a singular perturbation problem with variable coefficients in two dimensions, J. Sci. Comput. 41 (2009), no. 2, 200–220. MR 2550367, DOI 10.1007/s10915-009-9292-2
Houde Han and Zhongyi Huang, Tailored finite point method for steady-state reaction-diffusion equations, Commun. Math. Sci. 8 (2010), no. 4, 887–899. MR 2744911
Houde Han, Zhongyi Huang, and R. Bruce Kellogg, A tailored finite point method for a singular perturbation problem on an unbounded domain, J. Sci. Comput. 36 (2008), no. 2, 243–261. MR 2434846, DOI 10.1007/s10915-008-9187-7
H. Han and R. B. Kellogg, Differentiability properties of solutions of the equation $-\epsilon ^2\Delta u+ru=f(x,y)$ in a square, SIAM J. Math. Anal. 21 (1990), no. 2, 394–408. MR 1038899, DOI 10.1137/0521022
Zhongyi Huang, Tailored finite point method for the interface problem, Netw. Heterog. Media 4 (2009), no. 1, 91–106. MR 2480424, DOI 10.3934/nhm.2009.4.91
A. M. Il′in, A difference scheme for a differential equation with a small parameter multiplying the highest derivative, Mat. Zametki 6 (1969), 237–248 (Russian). MR 260195
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
Jichun Li and Yitung Chen, Uniform convergence analysis for singularly perturbed elliptic problems with parabolic layers, Numer. Math. Theory Methods Appl. 1 (2008), no. 2, 138–149. MR 2441969
Jichun Li and I. M. Navon, Uniformly convergent finite element methods for singularly perturbed elliptic boundary value problems: convection-diffusion type, Comput. Methods Appl. Mech. Engrg. 162 (1998), no. 1-4, 49–78. MR 1645349, DOI 10.1016/S0045-7825(97)00329-0
Zi-Cai Li, Heng-Shuing Tsai, Song Wang, and John J. H. Miller, Accurate and approximate analytic solutions of singularly perturbed differential equations with two-dimensional boundary layers, Comput. Math. Appl. 55 (2008), no. 11, 2602–2622. MR 2416029, DOI 10.1016/j.camwa.2007.10.027
Jens M. Melenk, $hp$-finite element methods for singular perturbations, Lecture Notes in Mathematics, vol. 1796, Springer-Verlag, Berlin, 2002. MR 1939620, DOI 10.1007/b84212
John J. H. Miller, On the convergence, uniformly in $\varepsilon$, of difference schemes for a two point boundary singular perturbation problem, Numerical analysis of singular perturbation problems (Proc. Conf., Math. Inst., Catholic Univ., Nijmegen, 1978) Academic Press, London-New York, 1979, pp. 467–474. MR 556537
K. W. Morton, Numerical solution of convection-diffusion problems, Applied Mathematics and Mathematical Computation, vol. 12, Chapman & Hall, London, 1996. MR 1445295
E. O’Riordan and G. I. Shishkin, Parameter uniform numerical methods for singularly perturbed elliptic problems with parabolic boundary layers, Appl. Numer. Math. 58 (2008), no. 12, 1761–1772. MR 2464809, DOI 10.1016/j.apnum.2007.11.003
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, A finite difference scheme on a priori adapted meshes for a singularly perturbed parabolic convection-diffusion equation, Numer. Math. Theory Methods Appl. 1 (2008), no. 2, 214–234. MR 2441974
Martin Stynes, Steady-state convection-diffusion problems, Acta Numer. 14 (2005), 445–508. MR 2170509, DOI 10.1017/S0962492904000261