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

Authors:
Houde Han and Zhongyi Huang

Journal:
Math. Comp. **82** (2013), 213-226

MSC (2010):
Primary 65-xx; Secondary 35-xx

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

Published electronically:
June 6, 2012

MathSciNet review:
2983022

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 for some cases without any prior knowledge of the boundary layers. Our numerical examples show the efficiency and reliability of our method.

**1.**M. Abramowitz and I. A. Stegun,*Handbook of mathematical functions*, National Bureau of Standards, 1964.**2.**A.E. Berger, H.D. Han and R.B. Kellogg,*A priori estimates and analysis of a numerical method for a turning point problem*, Math. Comp.,**42**(1984): 465-492. MR**736447 (85h:65158)****3.**I. Brayanov and I. Dimitrova,*Uniformly convergent high-order schemes for a 2D elliptic reaction-diffusion problem with anisotropic coefficients*, Lecture Notes In Computer Science**2542**(2003): 395-402. MR**2053277****4.**L.C. Evans,*Partial differential equations*, American Mathematical Society, Providence, RI, 1998. MR**1625845 (99e:35001)****5.**I. S. Gradshteyn and I. M. Ryzhik,*Table of integrals, series and products*, 6th Ed., Academic Press, 2000. MR**1773820 (2001c:00002)****6.**H. Han and Z. Huang,*A tailored finite point method for the Helmholtz equation with high wave numbers in heterogeneous medium*, J. Comp. Math,**26**(2008): 728-739. MR**2444729 (2009j:65156)****7.**H. Han and Z. Huang,*Tailored finite point method for a singular perturbation problem with variable coefficients in two dimensions*, J. Sci. Comp.,**41**(2009): 200-220. MR**2550367 (2010m:65249)****8.**H. Han and Z. Huang,*Tailored finite point method for steady-state reaction-diffusion equations*, Commun. Math. Sci.**8**(2010), no. 4, 887-899. MR**2744911 (2012a:65332)****9.**H. Han, Z. Huang and B. Kellogg,*A Tailored finite point method for a singular perturbation problem on an unbounded domain*, J. Sci. Comp.**36**(2008): 243-261. MR**2434846 (2010c:65192)****10.**H. Han and R. B. Kellogg,*Differentiability properties of solutions of the equation in a square*, SIAM J. Math. Anal.**21**(1990): 394-408 . MR**1038899 (91e:35025)****11.**Z. Huang,*Tailored finite point method for the interface problem*, Networks and Heterogeneous Media,**4**(2009): 91-106. MR**2480424 (2010f:65223)****12.**A.M. Il'in,*Differencing scheme for a differential equation with a small parameter affecting the highest derivative*, Math. Notes,**6**(1969): 596-602. MR**0260195 (41:4823)****13.**R. B. Kellogg and M. Stynes,*Sharpened bounds for corner singularities and boundary layers in a simple convection-diffusion problem*, Applied Mathematics Letters**20**(2007): 539-544. MR**2303990 (2007k:35093)****14.**J. Li and Y. Chen,*Uniform convergence analysis for singularly perturbed elliptic problems with parabolic layers*, Numer. Math. Theor. Meth. Appl.**1**(2008): 138-149. MR**2441969 (2009e:65180)****15.**J. Li and I.M. Navon,*Uniformly convergent finite element methods for singularly perturbed elliptic boundary value problems: convection-diffusion*, Comput. Methods Appl. Mech. Engrg.**162**(1998): 49-78. MR**1645349 (99f:65172)****16.**Z.C. Li, H.S. Tsai, S. Wang and J.J.H. Miller,*Accurate and approximate analytic solutions of singularly perturbed differential equations with two-dimensional boundary layers*, Comput. Math. Appl.,**55**(2008): 2602-2622. MR**2416029 (2009e:65106)****17.**J.M. Melenk,*hp-finite element methods for singular perturbations. Lecture Notes in Mathematics*, 1796. Springer-Verlag, Berlin, 2002. MR**1939620 (2003i:65108)****18.**J.J.H. Miller,*On the convergence, uniformly in , of difference schemes for a two-point boundary singular perturbation problem*, Numerical analysis of singular perturbation problems, eds., Hernker P. W. and Miller J. J, H., Academic Press, 467-474, 1979. MR**556537 (81f:65061)****19.**K.W. Morton,*Numerical solution of converction-diffusion problems*, Applied Mathematics and Mathematical Computation**12**, Chapman and Hall, London, 1996. MR**1445295 (98b:65004)****20.**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): 1761-1772. MR**2464809 (2009j:65366)****21.**H.-G. Roos, M. Stynes and L. Tobiska,*Robust numerical methods for singularly perturbed differential equations*, Springer, Berlin, 2nd ed., 2008. MR**2454024 (2009f:65002)****22.**G. I. Shishkin,*A finite difference scheme on a priori adapted meshes for a singularly perturbed parabolic convection-diffusion equation*, Numer. Math. Theor. Meth. Appl.**1**(2008): 214-234. MR**2441974 (2009e:65124)****23.**M. Stynes,*Steady-state convection-diffusion problems*, Acta Numerica,**14**(2005): 445-508. MR**2170509 (2007d:35050)**

Retrieve articles in *Mathematics of Computation*
with MSC (2010):
65-xx,
35-xx

Retrieve articles in all journals with MSC (2010): 65-xx, 35-xx

Additional Information

**Houde Han**

Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Email:
hhan@math.tsinghua.edu.cn

**Zhongyi Huang**

Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Email:
zhuang@math.tsinghua.edu.cn

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

Keywords:
Tailored finite point method,
singular perturbation problem,
boundary layer,
discrete maximum principle

Received by editor(s):
November 29, 2009

Received by editor(s) in revised form:
May 2, 2010

Published electronically:
June 6, 2012

Additional Notes:
H. Han was supported by the NSFC Project No. 10971116.

Z. Huang was supported by the NSFC Project No. 11071139, the National Basic Research Program of China under the grant 2011CB309705, Tsinghua University Initiative Scientific Research Program.

Article copyright:
© Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.