On the numerical approximations of stiff convection–diffusion equations in a circle

Abstract

Our aim in this article is to study the numerical solutions of singularly perturbed convection–diffusion problems in a circular domain and provide as well approximation schemes, error estimates and numerical simulations. To resolve the oscillations of classical numerical solutions due to the stiffness of our problem, we construct, via boundary layer analysis, the so-called boundary layer elements which absorb the boundary layer singularities. Using a \(P_1\) classical finite element space enriched with the boundary layer elements, we obtain an accurate numerical scheme in a quasi-uniform mesh.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

References

  1. 1.

    Andronov, I., Bouche, D., Molinet, F.: Asymptotic and hybrid methods in electromagnetics. IEE Electromagn. Waves Ser., vol. 48 (2005)

  2. 2.

    Bardos, C.: Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d’approximation; application à l’équation de transport, Ann. Sci. École Norm. Sup. r. (4) 3, 185–233 (1970) (in French)

  3. 3.

    Brooks, A., Hughes, T.: Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Method Appl. Mech. Eng. 32, 199–259 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  4. 4.

    Chipot, M., Correa, F.J.S.A.: Boundary layer solutions to functional elliptic equations. Bull. Braz. Math. Soc. 40, 381–393 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. 5.

    Chipot, M., Guesmia, S.: Correctors for some asymptotic problems. Tr. Mat. Inst. Steklova 270, 266–280 (2010)

    MathSciNet  Google Scholar 

  6. 6.

    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, New York (1973)

    Google Scholar 

  7. 7.

    Eckhaus, W., de Jager, E.M.: Asymptotic solutions of singular perturbation problems for linear differential equations of elliptic type. Arch. Rational Mech. Anal. 23, 26–86 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  8. 8.

    Eckhaus, W.: Boundary layers in linear elliptic singular perturbations. SIAM Rev. 14, 225–270 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  9. 9.

    Weinan, E.: Boundary layer theory and the zero viscosity limit of the NavierStokes equation. Acta Math. Sin. (Engl. Series) 16, 207–218 (2000)

    Google Scholar 

  10. 10.

    García-Archilla, B.: Shishkin mesh simulation: a new stabilization technique for convection-diffusion problems. Comput. Methods Appl. Mech. Eng. 256, 1–16 (2013)

    Article  Google Scholar 

  11. 11.

    Grasman, J.: On the birth of boundary layers. In: Math. Centre Tracts, vol. 36. Mathematisch Centrum, Amsterdam (1971)

  12. 12.

    Grasman, J.: An elliptic singular perturbation problem with almost characteristic boundaries. J. Math. Anal. Appl. 46, 438–446 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  13. 13.

    Hemker, P.W.: A singularly perturbed model problem for numerical computation. J. Comput. Appl. Math. 76, 277–285 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  14. 14.

    Hughes, T.J.R.: The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Dover Publications, Mineola (2000)

    Google Scholar 

  15. 15.

    Hong, Y., Jung, C., Laminie, J.: Singularly perturbed reaction-diffusion equations in a circle with numerical applications. Int. J. Comput. Math. (2013, to appear)

  16. 16.

    John, V., Knobloch, P.: On spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations. Comput. Methods Appl. Mech. Eng. 196, 2197–2215 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  17. 17.

    John, V., Knobloch, P., Savescu, S.B.: A posteriori optimization of parameters in stabilized methods for convection-diffusion problems-Part I. Comput. Methods Appl. Mech. Eng. 200, 2916–2929 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  18. 18.

    Jung, C.-Y., Temam, R.: Interaction of boundary layers and corner singularities. Discrete Cont. Dyn. Syst. 23, 315–339 (2009)

    MATH  MathSciNet  Google Scholar 

  19. 19.

    Jung, C.-Y., Temam, R.: Finite volume approximation of one-dimensional stiff convection-diffusion equations. J. Sci. Comput. 41, 384–410 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  20. 20.

    Jung, C.-Y., Temam, R.: Finite volume approximation of two-dimensional stiff problems. Int. J. Numer. Anal. Model. 7, 462–476 (2010)

    MATH  MathSciNet  Google Scholar 

  21. 21.

    Jung, C.-Y., Temam, R.: Convection-diffusion equations in a circle: the compatible case. J. Math. Pures Appl. 96, 88–107 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  22. 22.

    Jung, C.-Y., Temam, R.: Singular perturbations and boundary layer theory for convection-diffusion equations in a circle: the generic noncompatible case. SIAM J. Math. Anal. 44, 4274–4296 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  23. 23.

    Jung, C.-Y., Petcu, M., Temam, R.: Singular perturbation analysis on a homogeneous ocean circulation model. Anal. Appl. (Singap.) 9, 275–313 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  24. 24.

    Knaub, K.R., O’Malley, R.E.: The motion of internal layers in singularly perturbed advection-diffusion-reaction equations. Stud. Appl. Math 112, 1–15 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  25. 25.

    Kellogg, R.B., Stynes, M.: Layers and corner singularities in singularly perturbed elliptic problems. BIT 48, 309–314 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  26. 26.

    Lions, J.L.: Perturbations Singulières dans les Problèmes aux Limites et en Contrôle Optimal. In: Lecture Notes in Math., vol. 323. Springer, Berlin (1973) (in French)

  27. 27.

    Nguyen, H., Gunzburger, M., Ju, L., Burkardt, J.: Adaptive anisotropic meshing for steady convection-dominated problems. Comput. Methods Appl. Mech. Eng. 198, 2964–2981 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  28. 28.

    Oleinik, O.A., Samokhin, V.N.: Mathematical models in boundary layer theory. In: Applied mathematics and mathematical computation. Chapman and Hall, Boca Raton (1999)

    Google Scholar 

  29. 29.

    O’Malley, R.E.: Singularly perturbed linear two-point boundary value problems. SIAM Rev. 50(3), 459–482 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  30. 30.

    Persson, P.O., Strang, G.: A simple mesh generator in MATLAB. SIAM Rev. 46, 329–345 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  31. 31.

    Prandtl, L.: Verber Flüssigkeiten bei sehr kleiner Reibung, Verk. III Intem. Math. Kongr. Heidelberg, pp. 484–491 (1905)

  32. 32.

    Roos, H.-G., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations. Springer, Berlin (2008)

    MATH  Google Scholar 

  33. 33.

    Strang, G., Fix, G.J.: An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs (1973)

    MATH  Google Scholar 

  34. 34.

    Shih, S., Kellogg, R.B.: Asymptotic analysis of a singular perturbation problem. Siam J. Math. Anal. 18, 1467–1511 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  35. 35.

    Stynes, M.: Steady-state convection-diffusion problems. Acta Numer. 14, 445–508 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  36. 36.

    Stynes, M., Tobiska, L.: The SDFEM for a convection-diffusion problem with a boundary layer: optimal error analysis and enhancement of accuracy. SIAM J. Numer. Anal. 41, 1620–1642 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  37. 37.

    Temam, R., Wang, X.: Boundary layers associated with incompressible Navier-Stokes equations: the noncharacteristic boundary case. J. Differ. Equ. 179, 647–686 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  38. 38.

    Temme, N.M.: Analytical methods for an elliptic singular perturbation problem in a circle. J. Comput. Appl. Math. 207, 301–322 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  39. 39.

    Vishik, M.I., Lyusternik, L.A.: Regular degeneration and boundary layer for linear differential equations with small parameter. Uspekhi Mat. Nauk. 12, 3–122 (1957)

    MATH  MathSciNet  Google Scholar 

  40. 40.

    Zhang, Z.: Finite element superconvergence on Shishkin mesh for 2-d convection-diffusion problems. Math. Comp. 72, 1147–1177 (2003)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Youngjoon Hong.

Additional information

This work was supported in part by NSF Grants DMS 0906440 and DMS 1206438, and by the Research Fund of Indiana University and by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2012R1A1B3001167).

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hong, Y., Jung, CY. & Temam, R. On the numerical approximations of stiff convection–diffusion equations in a circle. Numer. Math. 127, 291–313 (2014). https://doi.org/10.1007/s00211-013-0585-x

Download citation

Mathematics Subject Classification

  • 74S05
  • 34D15
  • 34E05
  • 76E06