Advertisement

Partial Differential Equations with Oscillatory Coefficients

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

Abstract

We present here some efficient techniques to approximate solutions of partial differential equations with oscillatory coefficients. We consider a simple one-dimensional case that still keeps most of the difficulties present in more sophisticated problems. We discuss three different approximation techniques: classical finite elements, homogenization, and Multiscale Finite Element methods (MsFEM). We show the advantages and pitfalls of each of the techniques, and present numerical results.

Keywords

Finite Element Method Asymptotic Expansion Element Method Discretization Finite Element Approximation Multiscale Method 
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.

References

  1. 8.
    Allaire, G. (2002). Shape optimization by the homogenization method. In Applied Mathematical Sciences (Vol. 146). New York: Springer. MR1859696 (2002h:49001).Google Scholar
  2. 24.
    Babuška, I., Caloz, G., & Osborn, J. E. (1994). Special finite element methods for a class of second order elliptic problems with rough coefficients. SIAM Journal of Numerical Analysis, 31(4), 945–981. MR1286212 (95g:65146).Google Scholar
  3. 25.
    Babuška, I., & Osborn, J. E. (1983). Generalized finite element methods: Their performance and their relation to mixed methods. SIAM Journal of Numerical Analysis, 20(3), 510–536. MR701094 (84h:65076).Google Scholar
  4. 26.
    Babuška I., & Osborn, J. E. (1985). Finite element methods for the solution of problems with rough input data (Oberwolfach, 1983). Lecture Notes in Mathematics (Vol. 1121, pp. 1–18). Berlin: Springer. MR806382 (86m:65138).Google Scholar
  5. 45.
    Burman, E., Guzman, J., Sanchez, M. A., & Sarkis, M. (2016). Robust flux error estimation of Nitsche’s method for high contrast interface problems. arXiv:1602.00603v1 [math.NA].Google Scholar
  6. 49.
    Chechkin, G. A., Piatnitski, A. L., & Shamaev, A. S. (2007). Homogenization: Methods and applications. Translations of Mathematical Monographs (Vol. 234). Providence, RI: American Mathematical Society. Translated from the 2007 Russian original by Tamara Rozhkovskaya. MR2337848.Google Scholar
  7. 56.
    Cioranescu, D., & Donato, P. (1999). An introduction to homogenization. Oxford Lecture Series in Mathematics and Its Applications (Vol. 17). New York: The Clarendon Press/Oxford University Press. MR1765047 (2001j:35019).Google Scholar
  8. 60.
    Coutinho, A. L. G. A., Dias, C. M., Alves, J. L. D., Landau, L., Loula, A. F. D., Malta, S. M. C., et al. (2004). Stabilized methods and post-processing techniques for miscible displacements. Computational Methods in Applied Mechanical Engineering, 193(15–16), 1421–1436. MR2068902 (2005b:76083).Google Scholar
  9. 67.
    de Groen, P. P. N., & Hemker, P. W. (1979). Error bounds for exponentially fitted Galerkin methods applied to stiff two-point boundary value problems. In Numerical analysis of singular perturbation problems (Proc. Conf., Math. Inst., Catholic Univ., Nijmegen, 1978) (pp. 217–249). London: Academic Press. MR556520 (81a:65076).Google Scholar
  10. 73.
    Efendiev, Y., Galvis, J., & Hou, T. Y. (2013). Generalized multiscale finite element methods (GMsFEM). Journal of Computational Physics, 251, 116–135. doi:10.1016/ j.jcp.2013.04.045. MR3094911.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 74.
    Efendiev, Y., Galvis, J., & Wu, X.-H. (2011). Multiscale finite element methods for high-contrast problems using local spectral basis functions. Journal of Computational Physics, 230(4), 937–955. doi:10.1016/j.jcp.2010.09.026. MR2753343.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 77.
    Efendiev, Y., Hou, T., & Ginting, V. (2004). Multiscale finite element methods for nonlinear problems and their applications. Communications in Mathematical Science, 2(4), 553–589. MR2119929 (2005m:65265).Google Scholar
  13. 78.
    Efendiev, Y. R., Hou, T. Y., & Wu, X.-H. (2000). Convergence of a nonconforming multiscale finite element method. SIAM Journal of Numerical Analysis, 37(3), 888–910 (electronic). MR1740386 (2002a:65176).Google Scholar
  14. 80.
    Efendiev, Y., & Pankov, A. (2003). Numerical homogenization of monotone elliptic operators. Multiscale Modelling and Simulation, 2(1), 62–79 (electronic). MR2044957 (2005a:65153).Google Scholar
  15. 81.
    Efendiev, Y. R., & Wu, X.-H. (2002). Multiscale finite element for problems with highly oscillatory coefficients. Numerical Mathematics, 90(3), 459–486. MR1884226 (2002m:65114).Google Scholar
  16. 95.
    Franca, L. P., Madureira, A. L., & Valentin, F. (2005).Towards multiscale functions: Enriching finite element spaces with local but not bubble-like functions. Computational Methods in Applied Mechanical Engineering, 194(27–29), 3006–3021. MR2142535 (2006a:65159).Google Scholar
  17. 105.
    Gilbarg, D., & Trudinger, N. S. (2001). Elliptic partial differential equations of second order. Classics in Mathematics. Berlin: Springer. Reprint of the 1998 edition. MR1814364 (2001k:35004).Google Scholar
  18. 111.
    Guzman, J., Sanchez, M. A., & Sarkis, M. (2016). A finite element method for high-contrast interface problems with error estimates independent of contrast. arXiv:1507.03873v2.Google Scholar
  19. 126.
    Holmes, M. H. (2013). Introduction to perturbation methods. Texts in Applied Mathematics (2nd ed., Vol. 20). New York: Springer. MR2987304.Google Scholar
  20. 127.
    Hou, T. Y. (2003). Numerical approximations to multiscale solutions in partial differential equations. In Frontiers in numerical analysis (Durham, 2002) (pp. 241–301). MR2006969 (2004m:65219).Google Scholar
  21. 128.
    Hou, T. Y., & Liu, P. (2016). Optimal local multi-scale basis functions for linear elliptic equations with rough coefficient. Discrete and Continuous Dynamical Systems, 36(8), 4451–4476. ISSN:1078-0947; doi:10.3934/dcds.2016.36.4451. MR3479521.Google Scholar
  22. 129.
    Hou, T. Y., & Wu, X.-H. (1997). A multiscale finite element method for elliptic problems in composite materials and porous media. Journal of Computational Physics, 134(1), 169–189. MR1455261 (98e:73132).Google Scholar
  23. 130.
    Hou, T. Y., Wu, X.-H., & Cai, Z. (1999). Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Mathematics of Computation, 68(227), 913–943. MR1642758 (99i:65126).Google Scholar
  24. 131.
    Hou, T. Y., Wu, X.-H., & Zhang, Y. (2004). Removing the cell resonance error in the multiscale finite element method via a Petrov-Galerkin formulation. Communications in Mathematical Science, 2(2), 185–205. MR2119937.Google Scholar
  25. 154.
    Målqvist, A. (2011). Multiscale methods for elliptic problems. Multiscale Modelling and Simulation, 9(3), 1064–1086. doi:10.1137/090775592. MR2831590 (2012j:65419).Google Scholar
  26. 155.
    Målqvist, A., & Peterseim, D. (2014). Localization of elliptic multiscale problems. Mathematics of Computation, 83(290), 2583–2603. doi:10.1090/S0025-5718-2014-02868-8. MR3246801.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 161.
    Ming, P., & Yue, X. (2006). Numerical methods for multiscale elliptic problems. Journal of Computational Physics, 214(1), 421–445. MR2208685 (2006j:65359).Google Scholar
  28. 162.
    Moskow, S., & Vogelius, M. (1997). First-order corrections to the homogenised eigenvalues of a periodic composite medium: A convergence proof. Proceedings of the Royal Society of Edinburgh Section A, 127(6), 1263–1299. MR1489436 (99g:35018).Google Scholar
  29. 167.
    Paredes, D., Valentin, F., & Versieux, H.M. (2017). On the robustness of multiscale hybrid-mixed methods. Mathematics of Computation, 86(304), 525–548. ISSN: 0025-5718; doi:10.1090/mcom/3108. MR3584539.Google Scholar
  30. 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
  31. 181.
    Sangalli, G. (2003). Capturing small scales in elliptic problems using a residual-free bubbles finite element method. Multiscale Modelling and Simulation, 1(3), 485–503 (electronic). MR2030161 (2004m:65202).Google Scholar
  32. 182.
    Sarkis, M., & Versieux, H. (2008). Convergence analysis for the numerical boundary corrector for elliptic equations with rapidly oscillating coefficients. SIAM Journal of Numerical Analysis, 46(2), 545–576. MR2383203.Google Scholar
  33. 189.
    Versieux, H. M., & Sarkis, M. (2006). Numerical boundary corrector for elliptic equations with rapidly oscillating periodic coefficients. Communications in Numerical Methods in Engineering, 22(6), 577–589. MR2235030 (2007d:65117).Google Scholar
  34. 190.
    Versieux, H., & Sarkis, M. (2007). A three-scale finite element method for elliptic equations with rapidly oscillating periodic coefficients. Domain decomposition methods in science and engineering XVI. Lecture Notes in Computational Science and Engineering (Vol. 55, pp. 763–770). Berlin: Springer. MR2334173.Google Scholar
  35. 192.
    Wang, W., Guzmán, J., & Shu, C.-W. (2011). The multiscale discontinuous Galerkin method for solving a class of second order elliptic problems with rough coefficients. International Journal of Numerical Analysis and Modeling, 8(1), 28–47. MR2740478 (2012a:65346).Google Scholar
  36. 193.
    Weinan E, & Engquist, B. (2003). Multiscale modeling and computation. Notices of the American Mathematical Society, 50(9), 1062–1070. MR2002752 (2004m:65163).Google Scholar
  37. 199.
    Weinan E, Ming, P., & Zhang, P. (2005). Analysis of the heterogeneous multiscale method for elliptic homogenization problems. Journal of the American Mathematical Society, 18(1), 121–156 (electronic). MR2114818 (2005k:65246).Google Scholar
  38. 201.
    Zienkiewicz, O. C. (1997). Trefftz type approximation and the generalized finite element method—history and development. Computer Assisted Mechanics and Engineering Sciences, 4, 305–316.zbMATHGoogle 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