One-Dimensional Singular Perturbed Problems

  • Alexandre L. Madureira
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)


In this chapter, we introduce a singular perturbed problem and a numerical difficulty associate with its discretization. We first consider the one-dimensional advective dominated advection-diffusion problem, both in terms of numerical solutions and its asymptotic expansion. We then consider a more general asymptotic expansion, including a reaction term in the equation and considering the situation when the coefficients might depend on x as well.


Boundary Layer Exact Solution Asymptotic Expansion Finite Element Approximation Finite Element Discretization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 33.
    Boyce, W. E., & DiPrima, R. C. (1965). Elementary differential equations and boundary value problems. New York/London/Sydney: Wiley. MR0179403 (31 #3651).Google Scholar
  2. 42.
    Brezzi, F., Franca, L. P., & Russo, A. (1998). Further considerations on residual-free bubbles for advective-diffusive equations. Computational Methods in Applied Mechanical Engineering, 166(1–2), 25–33. doi:10.1016/S0045-7825(98)00080-2. MR1660137 (99j:65197).Google Scholar
  3. 44.
    Brooks, A. N., & Hughes, T. J. R. (1982). Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Computational Methods in Applied Mechanical Engineering, 32(1–3), 199–259. FENOMECH ’81, Part I (Stuttgart, 1981). doi:10.1016/0045-7825(82)90071-8. MR679322 (83k:76005).Google Scholar
  4. 58.
    Cockburn, B., Dong, B., Guzmán, J., Restelli, M., & Sacco, R. (2009). A hybridizable discontinuous Galerkin method for steady-state convection-diffusion-reaction problems. SIAM Journal of Scientific Computing, 31(5), 3827–3846. doi:10.1137/080728810. MR2556564 (2010m:65216).Google Scholar
  5. 62.
    Cronin, J., & O’Malley, R. E. Jr. (Eds.). (1999). Analyzing multiscale phenomena using singular perturbation methods. Proceedings of Symposia in Applied Mathematics (Vol. 56). Providence, RI: American Mathematical Society. Dedicated to the memory of William A. Harris, Jr.; Papers from the American Mathematical Society Short Course held in Baltimore, MD, January 5–6, 1998. MR1722494.Google Scholar
  6. 86.
    Falk, R. S. (2008). Finite elements for the Reissner–Mindlin plate. In Mixed finite elements, compatibility conditions, and applications. Lecture Notes in Mathematics. New York: Springer.Google Scholar
  7. 91.
    Franca, L. P., Frey, S. L., & Hughes, T. J. R. (1992). Stabilized finite element methods, I: Application to the advective-diffusive model. Computational Methods in Applied Mechanical Engineering, 95(2), 253–276. doi:10.1016/0045-7825(92)90143-8. MR1155924 (92m:76089).Google Scholar
  8. 93.
    Franca, L. P., & Madureira, A. L. (1993). Element diameter free stability parameters for stabilized methods applied to fluids. Computational Methods in Applied Mechanical Engineering, 105(3), 395–403. doi:10.1016/0045-7825(93)90065-6. MR1224304 (94g:76033).Google Scholar
  9. 94.
    Franca, L. P., Madureira, A. L., Tobiska, L., & Valentin, F. (2005). Convergence analysis of a multiscale finite element method for singularly perturbed problems. Multiscale Modelling and Simulation, 4(3), 839–866 (electronic). MR2203943 (2006k:65316).Google Scholar
  10. 98.
    Franca, L. P., Ramalho, J. V. A., & Valentin, F. (2006). Enriched finite element methods for unsteady reaction-diffusion problems. Communications in Numerical Methods Engineering, 22(6), 619–625. doi:10.1002/cnm.838. MR2235032.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 103.
    Galeão, A. C., Almeida, R. C., Malta, S. M. C., & Loula, A. F. D. (2004). Finite element analysis of convection dominated reaction-diffusion problems. Applied Numerical Mathematics, 48(2), 205–222. doi:10.1016/j.apnum.2003.10.002. MR2029331 (2004k:65219).Google Scholar
  12. 114.
    Harari, I., & Hughes, T. J. R. (1994). Stabilized finite element methods for steady advection-diffusion with production. Computational Methods in Applied Mechanical Engineering, 115(1–2), 165–191. doi:10.1016/0045-7825(94)90193-7. MR1278815 (95a:76059).Google Scholar
  13. 117.
    Harder, C., Paredes, D., & Valentin, F. (2015). On a multiscale hybrid-mixed method for advective-reactive dominated problems with heterogeneous coefficients. Multiscale Modelling and Simulation, 13(2), 491–518. doi:10.1137/130938499. MR3336297.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 126.
    Holmes, M. H. (2013). Introduction to perturbation methods. Texts in Applied Mathematics (2nd ed., Vol. 20). New York: Springer. MR2987304.Google Scholar
  15. 132.
    Hughes, T. J. R. (1978). A simple scheme for developing ‘upwind’ finite elements. International Journal for Numerical Methods in Engineering, 12, 1359–1365.CrossRefzbMATHGoogle Scholar
  16. 136.
    Hughes, T. J. R., Franca, L. P., & Hulbert, G. M. (1989). A new finite element formulation for computational fluid dynamics, VIII: The Galerkin/least-squares method for advective-diffusive equations. Computational Methods in Applied Mechanical Engineering 73(2), 173–189. doi:10.1016/0045-7825(89)90111-4. MR1002621 (90h:76007).Google Scholar
  17. 138.
    Il\(^{{\prime}}\) in, A. M. (1992). Matching of asymptotic expansions of solutions of boundary value problems. Translations of Mathematical Monographs (Vol. 102). Providence, RI: American Mathematical Society. Translated from the Russian by V. Minachin [V. V. Minakhin]. MR1182791.Google Scholar
  18. 139.
    Johnson, C. (1987). Numerical solution of partial differential equations by the finite element method. Cambridge: Cambridge University Press. MR925005 (89b:65003a).Google Scholar
  19. 179.
    Roos, H.-G., Stynes, M., & Tobiska, L. (2008). Robust numerical methods for singularly perturbed differential equations. Convection-diffusion-reaction and flow problems. Springer Series in Computational Mathematics (2nd ed., Vol. 24). Berlin: Springer. MR2454024 (2009f:65002).Google Scholar
  20. 188.
    Verhulst, F. (2005). Methods and applications of singular perturbations. Boundary layers and multiple timescale dynamics. Texts in Applied Mathematics (Vol. 50). New York: Springer. MR2148856.Google Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  • Alexandre L. Madureira
    • 1
    • 2
  1. 1.Laboratório Nacional de Computação Científica-LNCCPetrópolisBrazil
  2. 2.Fundação Getúlio Vargas-FGVRio de JaneiroBrazil

Personalised recommendations