Advances in Computational Mathematics

, Volume 45, Issue 3, pp 1291–1327 | Cite as

High dimensional finite elements for time-space multiscale parabolic equations

  • Wee Chin Tan
  • Viet Ha HoangEmail author


The paper develops the essentially optimal sparse tensor product finite element method for a parabolic equation in a domain in \(\mathbb {R}^{d}\) which depends on a microscopic scale in space and a microscopic scale in time. We consider the critical self similar case which has the most interesting homogenization limit. We solve the high dimensional time-space multiscale homogenized equation, which provides the solution to the homogenized equation which describes the multiscale equation macroscopically, and the corrector which encodes the microscopic information. For obtaining an approximation within a prescribed accuracy, the method requires an essentially optimal number of degrees of freedom that is essentially equal to that for solving a macroscopic parabolic equation in a domain in \(\mathbb {R}^{d}\). A numerical corrector is deduced from the finite element solution. Numerical examples for one and two dimensional problems confirm the theoretical results. Although the theory is developed for problems with one spatial microscopic scale, we show numerically that the method is capable of solving problems with more than one spatial microscopic scale.


High dimensional finite elements Time-space multiscale parabolic equations Optimal complexity Numerical corrector 

Mathematics Subject Classification (2010)

35B27 65M12 65M60 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The research topic originates from a discussion with Professor Christoph Schwab, ETH, Zurich. The authors gratefully acknowledge a postgraduate scholarship of A*Star, Singapore, the AcRF Tier 1 grant 2016-T1-001-202 RG30/16, the Singapore A*Star SERC grant 122-PSF-0007, and the AcRF Tier 2 grant MOE 2013-T2-1-095 ARC 44/13.


  1. 1.
    Abdulle, A., E, W., Engquist, B., Vanden-Eijnden, E.: The heterogeneous multiscale method. Acta Numer. 21, 1–87 (2012)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Abdulle, A., Huber, M.E.: Finite element heterogeneous multiscale method for nonlinear monotone parabolic homogenization problems. ESAIM Math. Model. Numer Anal. 50(6), 1659–1697 (2016)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Abdulle, A., Vilmart, G.: Coupling heterogeneous multiscale FEM with Runge-Kutta methods for parabolic homogenization problems: a fully discrete spacetime analysis. Math. Models Methods Appl. Sci. 22(6), 1250002 (2012)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23(6), 1482–1518 (1992)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bensoussan, A., Lions, J.-L., Papanicolaou, G.: Asymptotic analysis for periodic structures, volume 5 of studies in mathematics and its applications. North-Holland Publishing Co., Amsterdam (1978)zbMATHGoogle Scholar
  6. 6.
    Bieri, M., Andreev, R., Schwab, C.: Sparse tensor discretization of elliptic SPDEs. SIAM J. Sci. Comput. 31(6), 4281–4304 (2009/10)Google Scholar
  7. 7.
    Bungartz, H.-J., Griebel, M.: Sparse grids. Acta Numer. 13, 147–269 (2004)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Chen, S., E, W., Shu, C.-W.: The heterogeneous multiscale method based on the discontinuous Galerkin method for hyperbolic and parabolic problems. Multiscale Model Simul. 3(4), 871–894 (2005)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Chu, V.T., Hoang, V.H.: High dimensional finite elements for multiscale Maxwell-type equations. IMA J. Numer. Anal. 38(1), 227–270 (2018)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Chung, E.T., Efendiev, Y., Leung, W.T., Ye, S.: Generalized multiscale finite element methods for space-time heterogeneous parabolic equations. Comput. Math Appl. 76(2), 419–437 (2018)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Cioranescu, D., Damlamian, A., Griso, G.: The periodic unfolding method in homogenization. SIAM J. Math. Anal. 40(4), 1585–1620 (2008)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Douglas, J. Jr., Dupont, T.: Galerkin methods for parabolic equations. SIAM J. Numer. Anal. 7, 575–626 (1970)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Efendiev, Y., Hou, T.Y.: Multiscale finite element methods: theory and applications. surveys and tutorials in the applied mathematical sciences. Springer (2009)Google Scholar
  14. 14.
    Efendiev, Y., Pankov, A.: Numerical homogenization of nonlinear random parabolic operators. Multiscale Model Simul. 2(2), 237–268 (2004)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Efendiev, Y., Galvis, J., Hou, T.Y.: Generalized multiscale finite element methods (GMsFEM). J. Comput. Phys. 251, 116–135 (2013)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Evans, L.C.: Partial differential equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence (1998)Google Scholar
  17. 17.
    Geng, J., Shen, Z.: Convergence rates in parabolic homogenization with time-dependent periodic coefficients. J. Funct Anal. 272(5), 2092–2113 (2017)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Griebel, M., Oswald, P.: Tensor product type subspace splittings and multilevel iterative methods for anisotropic problems. Adv. Comput. Math. 4(1), 171–206 (1995)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Grisvard, P.: Elliptic problems in nonsmooth domains, volume 69 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2011)Google Scholar
  20. 20.
    Hoang, V.H., Schwab, C.: High-dimensional finite elements for elliptic problems with multiple scales. Multiscale Model. Simul. 3(1), 168–194 (2004/05)Google Scholar
  21. 21.
    Holmbom, A., Svanstedt, N., Wellander, N.: Multiscale convergence and reiterated homogenization of parabolic problems. Appl. Math. 50(2), 131–151 (2005)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Hou, T.Y., Wu, X.H.: A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134(1), 169–189 (1997)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Kazeev, V., Oseledets, I., Rakhuba, M., Schwab, C.: QTT-finite-element approximation for multiscale problems I: model problems in one dimension. Adv. Comput. Math. 43(2), 411–442 (2017)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Lions, J.L., Magenes, E.: Non-homogeneous boundary value problems and applications, vol. I. Springer, Berlin (1972)zbMATHGoogle Scholar
  25. 25.
    Målqvist, A., Persson, A.: Multiscale techniques for parabolic equations. Numer. Math. 138(1), 191–217 (2018)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Målqvist, A., Peterseim, D.: Localization of elliptic multiscale problems. Math. Comp. 83(290), 2583–2603 (2014)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Ming, P., Zhang, P.: Analysis of the heterogeneous multiscale method for parabolic homogenization problems. Math Comp. 76(257), 153–177 (2007)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Nguetseng, G.: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20(3), 608–623 (1989)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Owhadi, H., Zhang, L.: Homogenization of parabolic equations with a continuum of space and time scales. SIAM J. Numer. Anal. 46(1), 1–36 (2008)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Tan, W.C., Hoang, V.H.: High dimensional finite element method for multiscale nonlinear monotone parabolic equations. J. Comput. Appl. Math. 345, 471–500 (2019)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Wloka, J.: Partial differential equations. Cambridge University Press, Cambridge (1987)zbMATHGoogle Scholar
  32. 32.
    Woukeng, J.L.: Periodic homogenization of nonlinear non-monotone parabolic operators with three time scales. Ann. Mat. Pura Appl. (4) 189(3), 357–379 (2010)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Xia, B., Hoang, V.H.: High dimensional finite elements for multiscale wave equations. Multiscale Model. Simul. 12(4), 1622–1666 (2014)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Xia, B., Hoang, V.H.: High-dimensional finite element method for multiscale linear elasticity. IMA J. Numer. Anal. 35(3), 1277–1314 (2015)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Xia, B., Hoang, V.H.: Sparse tensor finite elements for elastic wave equation with multiple scales. J. Comput. Appl. Math. 282, 179–214 (2015)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Division of Mathematical Sciences, School of Physical and Mathematical SciencesNanyang Technological UniversitySingaporeSingapore

Personalised recommendations