Journal of Scientific Computing

, Volume 70, Issue 1, pp 272–300 | Cite as

Uniform Convergent Tailored Finite Point Method for Advection–Diffusion Equation with Discontinuous, Anisotropic and Vanishing Diffusivity



We propose two tailored finite point methods for the advection–diffusion equation with anisotropic tensor diffusivity. The diffusion coefficient can be very small in one direction in some part of the domain and be discontinuous at the interfaces. When flows advect from the vanishing-diffusivity region towards the non-vanishing diffusivity region, standard numerical schemes tend to cause spurious oscillations or negative values. Our proposed schemes have uniform convergence in the vanishing diffusivity limit, even when the solution exhibits interface and boundary layers. When the diffusivity is along the coordinates, the positivity and maximum principle can be proved. We use the value as well as their derivatives at the grid points to construct the scheme for nonaligned case, which makes it can achieve good accuracy and convergence for the derivatives as well, even when there exhibit boundary or interface layers. Numerical experiments are presented to show the performance of the proposed scheme.


Anisotropic diffusion Tailored finite point method Interface layer Boundary layer Uniform convergence 


  1. 1.
    Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 742–760 (1982)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ashby, S.F., Bosl, W.J., Falgout, R.D., Smith, S.G., Tompson, A.F., Williams, T.J.: A numerical simulation of groundwater flow and contaminant transport on the CRAY T3D and C90 supercomputers. Int. J. High Perform. Comput. Appl. 13(1), 80–93 (1999)CrossRefGoogle Scholar
  3. 3.
    Ern, A., Stephansen, A.F., Zunino, P.: A discontinuous Galerkin method with weighted averages for advection–diffusion equations with locally small and anisotropic diffusivity. IMA J. Numer. Anal. 29, 235–256 (2009)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Han, H., Huang, Z., Kellogg, B.: A tailored finite point method for a singular perturbation problem on an unbounded domain. J. Sci. Comp. 36, 243–261 (2008)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Han, H., Huang, Z.Y.: Tailored finite point method for a singular perturbation problem with variable coefficients in two dimensions. J. Sci. Comput. 49, 200–220 (2009)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Han, H., Huang, Z.Y.: Tailored finite point method for steady-state reaction–diffusion equation. Commun. Math. Sci. 8, 887–899 (2010)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Han, H., Huang, Z.Y.: Tailored finite point method based on exponential bases for convection–diffusion–reaction equation. Math. Comput. 82, 213–226 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Han, H., Huang, Z.Y., Ying, W.J.: A semi-discrete tailored finite point method for a class of anisotropic diffusion problems. Comput. Math. Appl. 65, 1760–1774 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hyman, J., Morel, J., Shashkov, M., Steinberg, S.: Mimetic finite difference methods for diffusion equations. Comput. Geosci. 6, 333–352 (2002)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Herbin,R.,Hubert, F.: Benchmark on discretization schemes for anisotropic diffusion problems on general grids. In: Eymard, R., Herard, J.-M. (eds.) Finite Volumes for Complex Applications V, pp. 659–692. John Wiley & Sons (2008)Google Scholar
  11. 11.
    Kellogg, R.B., Houde, Han: Numerical analysis of singular perturbation problems. Hemisphere Publishing Corp, USA. Distributed outside North America by Springer-Verlag, pp. 323–335 (1983)Google Scholar
  12. 12.
    Lipnikov, K., Shashkov, M., Svyatskiy, D., Vassilevski, Yu.: Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes regimes. J. Comput. Phys. 227, 492–512 (2007)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Lipnikov, K., Svyatskiy, D., Vassilevski, Y.: A monotone finite volume method for advection diffusion equations on unstructured polygonal meshes. J. Comput. Phys. 229, 4017–4032 (2010)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Motzkin, T.S., Wasow, W.: On the approximation of linear elliptic differential equations by difference equations with positive coefficients. J. Math. Phys. 31, 253–259 (1953)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Leveque, R.J., Li, Z.L.: The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 31, 1019–1044 (1994)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Oberman, A.M.: Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton–Jacobi equations and free boundary problems. SIAM J. Numer. Anal. 44(2), 879–895 (2006)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Pasdunkorale, J., Turner, I.W.: A second order control-volume finite-element least-squares strategy for simulating diffusion in strongly anisotropic media. J. Comput. Math. 23, 1–16 (2005)MathSciNetMATHGoogle Scholar
  18. 18.
    Sheng, Z.Q., Yue, J.Y., Yuan, G.W.: Monotone finite volume schemes of nonequilibrium radiation diffusion equations on distorted meshes. SIAM J. Sci. Comput. 31(4), 2915–2934 (2009)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Treguier, A.M.: Modelisation numerique pour loceanographie physique. Ann. Math. Blaise Pascal 9(2), 345–361 (2002)MathSciNetCrossRefGoogle Scholar
  20. 20.
    van Esa, B., Koren, B., de Blank, H.J.: Finite-difference schemes for anisotropic diffusion. J. Comput. Phys. 272, 526–549 (2014)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Wu, J.M., Gao, Z.M.: Interpolation-based second-order monotone finite volume schemes for anisotropic diffusion equations on general grids. J. Comput. Phys. 274, 569–588 (2014)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Younes, A., Fontaine, V.: Efficiency of mixed hybrid finite element and multipoint flux approximation methods on quadrangular grids and highly anisotropic media. Int. J. Numer. Methods Eng. 76, 314–336 (2008)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Yuan, G.W., Sheng, Z.Q.: Monotone finite volume schemes for diffusion equations on polygonal meshes. J. Comput. Phys. 227(12), 6288–6312 (2008)MathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Mathematics and Institute of Natural Sciences, MOE-LSCShanghai Jiao Tong UniversityShanghaiChina
  2. 2.Department of Mathematics and Institute of Natural SciencesShanghai Jiao Tong UniversityShanghaiChina

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