Tailored Finite Point Method for Parabolic Problems
-
Zhongyi Huang
und
Yi Yang
Abstract
In this paper, we propose a class of new tailored finite point methods (TFPM) for the numerical solution of parabolic equations. Our finite point method has been tailored based on the local exponential basis functions. By the idea of our TFPM, we can recover all the traditional finite difference schemes. We can also construct some new TFPM schemes with better stability condition and accuracy. Furthermore, combining with the Shishkin mesh technique, we construct the uniformly convergent TFPM scheme for the convection-dominant convection-diffusion problem. Our numerical examples show the efficiency and reliability of TFPM.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11322113
Award Identifier / Grant number: 91330203
Funding statement: This work was partially supported by the NSFC projects 11322113 and 91330203, and the Sino-German Science Center (Grant ID 1228) on the occasion of the Chinese-German Workshop on Computational and Applied Mathematics in Augsburg 2015.
References
[1] Bakhvalov N. S., On the optimization of methods for solving boundary value problems with boundary layers, Zh. Vychisl. Mat. Mat. Fiz. 9 (1969), 841–859. 10.1016/0041-5553(69)90038-XSuche in Google Scholar
[2] Dunne R. K. and O’Riordan E., Interior layers arising in linear singularly perturbed differential equations with discontinuous coefficients, Proceedings of the Fourth International Conference on Finite Difference Methods: Theory and Applications (FDM ’06), Rousse University, Bulgaria (2007), 29–38. Suche in Google Scholar
[3] Farrell P. A., Hemker P. W. and Shishkin G. I., Discrete approximations for singularly perturbed boundary value problems with parabolic layers I, J. Comput. Math. 14 (1996), 71–97. Suche in Google Scholar
[4] Gracia J. L. and O’Riordan E., Numerical approximation of solution derivatives of singularly perturbed parabolic problems of convection-diffusion type, Math. Comp. 85 (2016), 581–599. 10.1090/mcom/2998Suche in Google Scholar
[5] Han H. and Huang Z., Tailored finite point method for a singular perturbation problem with variable coefficients in two dimensions, J. Sci. Comput. 41 (2009), 200–220. 10.1007/s10915-009-9292-2Suche in Google Scholar
[6] Han H. and Huang Z., Tailored finite point method for steady-state reaction-diffusion equation, Commun. Math. Sci. 8 (2010), 887–899. 10.4310/CMS.2010.v8.n4.a5Suche in Google Scholar
[7] Han H. and Huang Z., An equation decomposition method for the numerical solution of a fourth-order elliptic singular perturbation problem, Numer. Methods Partial Differential Equations 28 (2012), 942–953. 10.1002/num.20666Suche in Google Scholar
[8] Han H. and Huang Z., Tailored finite point method based on exponential bases for convection-diffusion-reaction equation, Math. Comp. 82 (2013), 213–226. 10.1090/S0025-5718-2012-02616-0Suche in Google Scholar
[9] Han H. and Huang Z., The Tailored finite point method, Comput. Methods Appl. Math. 14 (2014), 321–345. 10.1515/cmam-2014-0012Suche in Google Scholar
[10] Han H., Huang Z. and Kellogg B., A Tailored finite point method for a singular perturbation problem on an unbounded domain, J. Sci. Comput. 36 (2008), 243–261. 10.1007/s10915-008-9187-7Suche in Google Scholar
[11] Hemker P. W. and Shishkin G. I., Discrete approximation of singularly perturbed parabolic PDEs with a discontinuous initial condition, Comput. Fluid Dyn. J. 2 (1994), 375–392. Suche in Google Scholar
[12] Huang Z., Tailored finite point method for the interface problem, Netw. Heterog. Media 4 (2009), 91–106. 10.3934/nhm.2009.4.91Suche in Google Scholar
[13] Huang Z., Tailored finite point method for numerical simulation of partial differential equations, Fifth international Congress of Chinese Mathematicians. Part 2 (ICCM ’10), AMS/IP Stud. Adv. Math. 51, American Mathematical Society, Providence (2012), 593–608. 10.1090/amsip/051.2/07Suche in Google Scholar
[14] Huang Z. and Yang X., Tailored finite point method for first order wave equation, J. Sci. Comput. 49 (2011), 351–366. 10.1007/s10915-011-9468-4Suche in Google Scholar
[15] Kopteva N., Uniform pointwise convergence of difference schemes for convection-diffusion problems on layer-adapted meshes, Computing 66 (2001), 179–197. 10.1007/s006070170034Suche in Google Scholar
[16] Kopteva N. and O’Riordan E., Shishkin meshes in the numerical solution of singularly perturbed differential equations, Int. J. Numer. Anal. Model. 7 (2010), 393–415. Suche in Google Scholar
[17] Miller J. J. H., O’Riordan E. and Shishkin G. I., Fitted mesh methods for problems with parabolic boundary layers, Math. Proc. R. Ir. Acad. 98 (1998), 173–190. Suche in Google Scholar
[18] 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. Suche in Google Scholar
[19] Stoyan G., Monotonic mesh approximation for a partial differential equation with one space variable, Differ. Equ. 18 (1983), 886–897. Suche in Google Scholar
[20] Strikwerda J. C., Finite Difference Schemes and Partial Differential Equations, SIAM, Philadelphia, 2004. 10.1137/1.9780898717938Suche in Google Scholar
[21] Titov V. A. and Shishkin G. I., A numerical solution of a parabolic equation with small parameters multiplying the derivatives with respect to the space variables, Tr. Inst. Mat. i Meh. Ural. Naucu Centr. Akad. Nauk SSR 21 (1976), 38–43. Suche in Google Scholar
© 2016 by De Gruyter
Artikel in diesem Heft
- Frontmatter
- Numerical Approximation of Multi-Phase Penrose–Fife Systems
- Tailored Finite Point Method for Parabolic Problems
- A Weakly Penalized Discontinuous Galerkin Method for Radiation in Dense, Scattering Media
- Robust Numerical Upscaling of Elliptic Multiscale Problems at High Contrast
- Chinese–German Computational and Applied Mathematics
- Functional A Posteriori Error Control for Conforming Mixed Approximations of Coercive Problems with Lower Order Terms
- Super-Exponentially Convergent Parallel Algorithm for Eigenvalue Problems with Fractional Derivatives
- A Nonconforming Finite Element Approximation for Optimal Control of an Obstacle Problem
- Construction of Scalar and Vector Finite Element Families on Polygonal and Polyhedral Meshes
- Optimal Control for the Thin Film Equation: Convergence of a Multi-Parameter Approach to Track State Constraints Avoiding Degeneracies
Artikel in diesem Heft
- Frontmatter
- Numerical Approximation of Multi-Phase Penrose–Fife Systems
- Tailored Finite Point Method for Parabolic Problems
- A Weakly Penalized Discontinuous Galerkin Method for Radiation in Dense, Scattering Media
- Robust Numerical Upscaling of Elliptic Multiscale Problems at High Contrast
- Chinese–German Computational and Applied Mathematics
- Functional A Posteriori Error Control for Conforming Mixed Approximations of Coercive Problems with Lower Order Terms
- Super-Exponentially Convergent Parallel Algorithm for Eigenvalue Problems with Fractional Derivatives
- A Nonconforming Finite Element Approximation for Optimal Control of an Obstacle Problem
- Construction of Scalar and Vector Finite Element Families on Polygonal and Polyhedral Meshes
- Optimal Control for the Thin Film Equation: Convergence of a Multi-Parameter Approach to Track State Constraints Avoiding Degeneracies
