Skip to main content
SpringerLink
Log in

Essentially optimal finite elements for multiscale elliptic eigenvalue problems

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

We consider a multiscale elliptic eigenvalue problem that depends on n separable microscopic scales. Using multiscale homogenization, we derive the multiscale homogenized eigenvalue problem whose solution contains all the possible eigenvalues and eigenfunctions of the homogenized eigenvalue problem. We develop the sparse tensor product finite element (FE) method for solving this multiscale homogenized problem, thus bypassing the expensive task of forming the homogenized equation. Although the multiscale homogenized eigenvalue problem is posed in a high dimensional tensorized domain, the sparse tensor product FEs achieve a prescribed level of accuracy requiring an essentially optimal number of degrees of freedom which differs from the optimal one by only a possible logarithmic multiplying factor, given that the solution to this problem is sufficiently regular. We show that the regularity requirement is achieved under the Lipschitz condition on the multiscale coefficient. Numerical examples on two- and three-scale eigenvalue problems support the theoretical error estimates.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abdulle, A., Arjmand, D., Paganoni, E.: Exponential decay of the resonance error in numerical homogenization via parabolic and elliptic cell problems. C. R. Math. Acad. Sci. Paris 357(6), 545–551 (2019)

    Article  MathSciNet  Google Scholar 

  2. Abdulle, A., E, W., Engquist, B., Vanden-Eijnden, E.: The heterogeneous multiscale method. Acta Numer. 21, 1–87 (2012)

    Article  MathSciNet  Google Scholar 

  3. Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23(6), 1482–1518 (1992)

    MathSciNet  MATH  Google Scholar 

  4. Allaire, G., Briane, M.: Multiscale convergence and reiterated homogenisation. Proc. Roy. Soc. Edinburgh Sect. A 126(2), 297–342 (1996)

    Article  MathSciNet  Google Scholar 

  5. Brown, D.L., Efendiev, Y., Hoang, V.H.: An efficient hierarchical multiscale finite element method for Stokes equations in slowly varying media. Multiscale Model. Simul. 11(1), 30–58 (2013)

    Article  MathSciNet  Google Scholar 

  6. Bungartz, H.-J., Griebel, M.: Sparse grids. Acta Numer. 13, 147–269 (2004)

    Article  MathSciNet  Google Scholar 

  7. Chen, Z., Zou, J.: Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math. 79(2), 175–202 (1998)

    Article  MathSciNet  Google Scholar 

  8. Chu, V.T., Hoang, V.H.: High dimensional finite elements for multiscale Maxwell equations. IMA J. Numer Anal. 38, 227–270 (2018)

    Article  MathSciNet  Google Scholar 

  9. Chu, V.T., Hoang, V.H.: High dimensional finite elements for two-scale Maxwell wave equations. J. Comput. Appl Math. 375(32), 112756 (2020)

    Article  MathSciNet  Google Scholar 

  10. Ciarlet, P.G.: The finite element method for elliptic problems, volume 40 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia. Reprint of the 1978 original [North-Holland, Amsterdam] (2002)

  11. Courant, R., Hilbert, D.: Methods of Mathematical Physics: Partial Differential Equations interscience (1962)

  12. E, W., Engquist, B.: The heterogeneous multiscale methods. Commun. Math. Sci. 1(1), 87–132 (2003)

    Article  MathSciNet  Google Scholar 

  13. Efendiev, Y., Galvis, J., Hou, T.Y.: Generalized multiscale finite element methods (GMsFEM). J. Comput. Phys. 251, 116–135 (2013)

    Article  MathSciNet  Google Scholar 

  14. Efendiev, Y., Hou, T.Y.: Multiscale Finite Element Methods: Theory and Applications. Surveys and Tutorials in the Applied Mathematical Sciences. Springer (2009)

  15. Hoang, V. H., Schwab, Ch.: Analytic regularity and polynomial approximation of stochastic, parametric elliptic multiscale PDEs. Anal. Appl. 11(01):1350001 (2013)

  16. Hoang, V.H.: Sparse tensor finite element method for periodic multiscale nonlinear monotone problems. Multiscale Model. Simul. 7(3), 1042–1072 (2008)

    Article  MathSciNet  Google Scholar 

  17. Hoang, V.H., Schwab, Ch.: High-dimensional finite elements for elliptic problems with multiple scales. Multiscale Model. Simul. 3(1):168–194 (2004/05)

  18. Hou, T.Y., De, H, Ka, C.L., Zhang, Z.: A fast hierarchically preconditioned eigensolver based on multiresolution matrix decomposition. Multiscale Model. Simul. 17(1), 260–306 (2019)

    Article  MathSciNet  Google Scholar 

  19. 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)

    Article  MathSciNet  Google Scholar 

  20. Isaacson, E., Keller, H.B.: Analysis of numerical methods. Dover Publications, Inc., New York. Corrected reprint of the 1966 original [Wiley, New York] (1994)

  21. Kesavan, S.: Homogenization of elliptic eigenvalue problems: part 1. Appl. Math. Optim. 5(1), 153–167 (1979)

    Article  MathSciNet  Google Scholar 

  22. Kirsch, A.: An introduction to the mathematical theory of inverse problems, vol. 120. Springer (2011)

  23. Målqvist, A., Peterseim, D.: Localization of elliptic multiscale problems. Math. Comp. 83(290), 2583–2603 (2014)

    Article  MathSciNet  Google Scholar 

  24. Målqvist, A., Peterseim, D.: Computation of eigenvalues by numerical upscaling. Numer. Math. 130(2), 337–361 (2015)

    Article  MathSciNet  Google Scholar 

  25. Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press (2003)

  26. Nguetseng, G.: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20(3), 608–623 (1989)

    Article  MathSciNet  Google Scholar 

  27. Osborn, J.E.: Spectral approximation for compact operators. Math. Comput. 29, 712–725 (1975)

    Article  MathSciNet  Google Scholar 

  28. Park, J.S.R., Hoang, V.H.: Hierarchical multiscale finite element method for multi-continuum media. J. Comput. Appl. Math. 369(20), 112588 (2020)

    Article  MathSciNet  Google Scholar 

  29. Schwab, Ch.: High dimensional finite elements for elliptic problems with multiple scales and stochastic data. In: Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), pp. 727–734, Higher Ed. Press, Beijing (2002)

  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)

    Article  MathSciNet  Google Scholar 

  31. Tan, W.C., Hoang, V.H.: Sparse tensor product finite element method for nonlinear multiscale variational inequalities of monotone type. IMA J. Numer. Anal. 40(3), 1875–1907 (2020)

    Article  MathSciNet  Google Scholar 

  32. Xia, B.X, Hoang, V.H.: High-dimensional finite element method for multiscale linear elasticity. IMA J. Numer. Anal. 35(3), 1277–1314 (2015)

    Article  MathSciNet  Google Scholar 

  33. Xia, B.X., Hoang, V.H.: Sparse tensor finite elements for elastic wave equation with multiple scales. J. Comput. Appl. Math. 282:179–214 (2015)

  34. Xie, H., Zhang, L., Owhadi, H.: Fast eigenpairs computation with operator adapted wavelets and hierarchical subspace correction. SIAM J. Numer. Anal. 57(6), 2519–2550 (2019)

    Article  MathSciNet  Google Scholar 

Download references

Funding

The research was financially supported by the Singapore A*Star SERC grant 122-PSF-0007 and the Singapore MOE Tier 2 grant MOE2017-T2-2-144.

Author information

Authors and Affiliations

  1. Department of Mathematics, The University of Danang - University of Science and Education, 459 Ton Duc Thang, Danang, Vietnam

    Pham Quy Muoi

  2. Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore, 637371, Singapore

    Wee Chin Tan & Viet Ha Hoang

Authors
  1. Pham Quy Muoi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wee Chin Tan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Viet Ha Hoang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Viet Ha Hoang.

Additional information

Communicated by: Lothar Reichel

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Muoi, P.Q., Tan, W.C. & Hoang, V.H. Essentially optimal finite elements for multiscale elliptic eigenvalue problems. Adv Comput Math 47, 80 (2021). https://doi.org/10.1007/s10444-021-09903-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10444-021-09903-5

Keywords

Mathematics Subject Classification (2010)

Search

Navigation