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Networks and Heterogeneous Media

2009, Volume 4, Issue 1: 91-106. doi: 10.3934/nhm.2009.4.91
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Tailored finite point method for the interface problem

  • 1.
    Dept. of Mathematical Sciences, Tsinghua University, Beijing 100084
  • Received: 01 July 2008 Revised: 01 November 2008

  • Primary: 65N22, 65N35; Secondary: 35J25.

  • In this paper, we propose a tailored-finite-point method for a numerical simulation of the second order elliptic equation with discontinuous coefficients. Our finite point method has been tailored to some particular properties of the problem, then we can get the approximate solution with the same behaviors as that of the exact solution very naturally. Especially, in one-dimensional case, when the coefficients are piecewise linear functions, we can get the exact solution with only one point in each subdomain. Furthermore, the stability analysis and the uniform convergence analysis in the energy norm are proved. On the other hand, our computational complexity is only \O(N) for N discrete points. We also extend our method to two-dimensional problems.

    Citation: Zhongyi Huang. Tailored finite point method for the interface problem[J]. Networks and Heterogeneous Media, 2009, 4(1): 91-106. doi: 10.3934/nhm.2009.4.91

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  • In this paper, we propose a tailored-finite-point method for a numerical simulation of the second order elliptic equation with discontinuous coefficients. Our finite point method has been tailored to some particular properties of the problem, then we can get the approximate solution with the same behaviors as that of the exact solution very naturally. Especially, in one-dimensional case, when the coefficients are piecewise linear functions, we can get the exact solution with only one point in each subdomain. Furthermore, the stability analysis and the uniform convergence analysis in the energy norm are proved. On the other hand, our computational complexity is only \O(N) for N discrete points. We also extend our method to two-dimensional problems.


  • This article has been cited by:

    1. Houde Han, Zhongyi Huang, Tailored Finite Point Method for a Singular Perturbation Problem with Variable Coefficients in Two Dimensions, 2009, 41, 0885-7474, 200, 10.1007/s10915-009-9292-2
    2. Feng Xie, An interface problem with singular perturbation on a subinterval, 2014, 2014, 1687-2770, 10.1186/s13661-014-0201-8
    3. Xiaomin Liu, Muhammad Abbas, Honghong Yang, Xinqiang Qin, Tahir Nazir, Novel finite point approach for solving time-fractional convection-dominated diffusion equations, 2021, 2021, 1687-1847, 10.1186/s13662-020-03178-8
    4. Zhongyi Huang, Xu Yang, Tailored Finite Point Method for First Order Wave Equation, 2011, 49, 0885-7474, 351, 10.1007/s10915-011-9468-4
    5. Yaning Xie, Zhongyi Huang, Wenjun Ying, A Cartesian grid based tailored finite point method for reaction-diffusion equation on complex domains, 2021, 97, 08981221, 298, 10.1016/j.camwa.2021.05.020
    6. Wang Kong, Zhongyi Huang, Asymptotic Analysis and Numerical Method for Singularly Perturbed Eigenvalue Problems, 2018, 40, 1064-8275, A3293, 10.1137/17M1136249
    7. Houde Han, Zhongyi Huang, An equation decomposition method for the numerical solution of a fourth-order elliptic singular perturbation problem, 2012, 28, 0749159X, 942, 10.1002/num.20666
    8. Houde Han, Zhongyi Huang, Wenjun Ying, A semi-discrete tailored finite point method for a class of anisotropic diffusion problems, 2013, 65, 08981221, 1760, 10.1016/j.camwa.2013.03.017
    9. Mehdi Dehghan, Rezvan Salehi, A meshfree weak-strong (MWS) form method for the unsteady magnetohydrodynamic (MHD) flow in pipe with arbitrary wall conductivity, 2013, 52, 0178-7675, 1445, 10.1007/s00466-013-0886-z
    10. Hongxu Lin, Feng Xie, Singularly perturbed second order semilinear boundary value problems with interface conditions, 2015, 2015, 1687-2770, 10.1186/s13661-015-0309-5
    11. Zhongyi Huang, Yi Yang, Tailored Finite Point Method for Parabolic Problems, 2016, 16, 1609-9389, 543, 10.1515/cmam-2016-0017
    12. Zhongyi Huang, Ye Li, Monotone finite point method for non-equilibrium radiation diffusion equations, 2016, 56, 0006-3835, 659, 10.1007/s10543-015-0573-x
    13. Po-Wen Hsieh, Yintzer Shih, Suh-Yuh Yang, A Tailored Finite Point Method for Solving Steady MHD Duct Flow Problems with Boundary Layers, 2011, 10, 1815-2406, 161, 10.4208/cicp.070110.020710a
    14. Dongdong Nie, Feng Xie, Singularly perturbed semilinear elliptic boundary value problems with discontinuous source term, 2016, 2016, 1687-2770, 10.1186/s13661-016-0673-9
    15. Houde Han, Zhongyi Huang, Shangyou Zhang, An iterative method based on equation decomposition for the fourth-order singular perturbation problem, 2013, 29, 0749159X, 961, 10.1002/num.21740
    16. V.P. Shyaman, A. Sreelakshmi, Ashish Awasthi, An adaptive tailored finite point method for the generalized Burgers’ equations, 2022, 62, 18777503, 101744, 10.1016/j.jocs.2022.101744
    17. V. P. Shyaman, A. Sreelakshmi, Ashish Awasthi, A higher order implicit adaptive finite point method for the Burgers' equation, 2023, 1023-6198, 1, 10.1080/10236198.2023.2197082
    18. V P Shyaman, A Sreelakshmi, Ashish Awasthi, An implicit tailored finite point method for the Burgers’ equation: leveraging the Cole-Hopf transformation, 2024, 99, 0031-8949, 075283, 10.1088/1402-4896/ad56d8
    19. V. P. Shyaman, A. Sreelakshmi, Ashish Awasthi, Streamlined numerical solutions of burgers' equations bridging the tailored finite point method and the Cole Hopf transformation, 2024, 101, 0020-7160, 789, 10.1080/00207160.2024.2384598
    20. A Sreelakshmi, V P Shyaman, Ashish Awasthi, An adaptive finite point scheme for the two-dimensional coupled burgers’ equation, 2024, 1017-1398, 10.1007/s11075-024-01936-3
    21. A. Sreelakshmi, V.P. Shyaman, Ashish Awasthi, An interwoven composite tailored finite point method for two dimensional unsteady Burgers' equation, 2024, 197, 01689274, 71, 10.1016/j.apnum.2023.11.007
    22. Ye Li, Ting Du, Zhongyi Huang, 2024, Chapter 5, 978-3-031-72331-5, 60, 10.1007/978-3-031-72332-2_5
    23. A Sreelakshmi, V P Shyaman, Ashish Awasthi, An integrated stairwise adaptive finite point scheme for the two-dimensional coupled Burgers’ equation, 2024, 56, 0169-5983, 065505, 10.1088/1873-7005/ad8d08
    24. V.P. Shyaman, A. Sreelakshmi, Ashish Awasthi, A higher order implicit stair-tailored scheme for the modified Burgers’ equation, 2025, 217, 03064549, 111284, 10.1016/j.anucene.2025.111284
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