Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Multiscale Finite Element Methods for an Elliptic Optimal Control Problem with Rough Coefficients

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We investigate multiscale finite element methods for an elliptic distributed optimal control problem with rough coefficients. They are based on the (local) orthogonal decomposition methodology of Målqvist and Peterseim.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

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

Similar content being viewed by others

Data Availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

References

  1. Abdulle, A.: On a priori error analysis of fully discrete heterogeneous multiscale FEM. Multiscale Model. Simul. 4, 447–459 (2005)

    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. Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. Academic Press, Amsterdam (2003)

    MATH  Google Scholar 

  4. Babuška, I.: The finite element method with Lagrange multipliers. Numer. Math. 20, 179–192 (1973)

    Article  Google Scholar 

  5. Babuška, I., Osborn, J.E.: Can a finite element method perform arbitrarily badly? Math. Comput. 69, 443–462 (2000)

    Article  MathSciNet  Google Scholar 

  6. Brenner, S.C., Garay, J.C., Sung, L.-Y.: Additive Schwarz preconditioners for a localized orthogonal decomposition method. Electron. Trans. Numer. Anal. 54, 234–255 (2021)

    Article  MathSciNet  Google Scholar 

  7. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York (2008)

    Book  Google Scholar 

  8. Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. RAIRO Anal. Numer. 8, 129–151 (1974)

    MathSciNet  MATH  Google Scholar 

  9. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  10. Dryja, M., Widlund, O.B.: An additive variant of the Schwarz alternating method in the case of many subregions. Technical Report 339. Department of Computer Science, Courant Institute (1987)

  11. Engquist, B., Li, X., Ren, W., Vanden-Eijnden, E.: The heterogeneous multiscale methods. Commun. Math. Sci. 1, 87–132 (2003)

    Article  MathSciNet  Google Scholar 

  12. Ela, W., Ming, P., Zhang, P.: Analysis of the heterogeneous multiscale method for elliptic homogenization problems. J. Am. Math. Soc. 18, 121–156 (2005)

    MathSciNet  MATH  Google Scholar 

  13. Efendiev, Y., Hou, T.Y.: Multiscale Finite Element Methods. Springer, New York (2009)

    MATH  Google Scholar 

  14. Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics, 2nd edn. Oxford University Press, Oxford (2014)

    Book  Google Scholar 

  15. Ge, L., Yan, N., Wang, L., Liu, W., Yang, D.: Heterogeneous multiscale method for optimal control problem governed by elliptic equations with highly oscillatory coefficients. J. Comput. Math. 36, 5 (2018)

    MathSciNet  MATH  Google Scholar 

  16. Greenbaum, A.: Iterative Methods for Solving Linear Systems. SIAM, Philadelphia (1997)

    Book  Google Scholar 

  17. Hellman, F., Målqvist, A.: Contrast independent localization of multiscale problems. Multiscale Model. Simul. 15, 1325–1355 (2017)

    Article  MathSciNet  Google Scholar 

  18. Henning, P., Målqvist, A.: Localized orthogonal decomposition techniques for boundary value problems. SIAM J. Sci. Comput. 36, A1609–A1634 (2014)

    Article  MathSciNet  Google Scholar 

  19. Henning, P., Ohlberger, M., Schweizer, B.: An adaptive multiscale finite element method. Multiscale Model. Simul. 12, 1078–1107 (2014)

    Article  MathSciNet  Google Scholar 

  20. Henning, P., Peterseim, D.: Oversampling for the multiscale finite element method. Multiscale Model. Simul. 11, 1149–1175 (2013)

    Article  MathSciNet  Google Scholar 

  21. Hetmaniuk, U., Klawonn, A.: Error estimates for a two-dimensional special finite element method based on component mode synthesis. Electron. Trans. Numer. Anal. 41, 109–132 (2014)

    MathSciNet  MATH  Google Scholar 

  22. Hetmaniuk, U.L., Lehoucq, R.B.: A special finite element method based on component mode synthesis. M2AN Math. Model. Numer. Anal. 44, 401–420 (2010)

    Article  MathSciNet  Google Scholar 

  23. Hou, T.Y., Wu, X.-H.: A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134, 169–189 (1997)

    Article  MathSciNet  Google Scholar 

  24. Hou, T.Y., Wu, X.-H., Cai, Z.: Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comput. 68, 913–943 (1999)

    Article  MathSciNet  Google Scholar 

  25. Hughes, T.J.R.: Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput. Methods Appl. Mech. Eng. 127, 387–401 (1995)

    Article  MathSciNet  Google Scholar 

  26. Hughes, T.J.R., Feijóo, G.R., Mazzei, L., Quincy, J.-B.: The variational multiscale method—a paradigm for computational mechanics. Comput. Methods Appl. Mech. Eng. 166, 3–24 (1998)

    Article  MathSciNet  Google Scholar 

  27. Hughes, T.J.R., Sangalli, G.: Variational multiscale analysis: the fine-scale Green’s function, projection, optimization, localization, and stabilized methods. SIAM J. Numer. Anal. 45, 539–557 (2007)

    Article  MathSciNet  Google Scholar 

  28. Kornhuber, R., Peterseim, D., Yserentant, H.: An analysis of a class of variational multiscale methods based on subspace decomposition. Math. Comput. 87, 2765–2774 (2018)

    Article  MathSciNet  Google Scholar 

  29. Lions, J.-L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, New York (1971)

    Book  Google Scholar 

  30. Målqvist, A., Persson, A., Stillfjord, T.: Multiscale differential Riccati equations for linear quadratic regulator problems. SIAM J. Sci. Comput. 40, A2406–A2426 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  32. Målqvist, A., Peterseim, D.: Numerical Homogenization by Localized Orthogonal Decomposition. SIAM, Philadelphia (2021)

    MATH  Google Scholar 

  33. Owhadi, H., Scovel, C.: Operator-Adapted Wavelets, Fast Solvers, and Numerical Homogenization. Cambridge University Press, Cambridge (2019)

    Book  Google Scholar 

  34. Peterseim, D., Scheichl, R.: Robust numerical upscaling of elliptic multiscale problems at high contrast. Comput. Methods Appl. Math. 16, 579–603 (2016)

    Article  MathSciNet  Google Scholar 

  35. Toselli, A., Widlund, O.B.: Domain Decomposition Methods—Algorithms and Theory. Springer, New York (2005)

    Book  Google Scholar 

  36. Zeng, J., Chen, Y., Liu, G.: Rough polyharmonic splines method for optimal control problem governed by parabolic systems with rough coefficient. Comput. Math. Appl. 80, 121–139 (2020)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

Portions of this research were conducted with high performance computing resources provided by Louisiana State University (http://www.hpc.lsu.edu).

Funding

Funding is provided by US National Science Foundation (Grant No. DMS-19-13035)

Author information

Authors and Affiliations

  1. Department of Mathematics and Center for Computation and Technology, Louisiana State University, Baton Rouge, LA, 70803, USA

    Susanne C. Brenner, José C. Garay & Li-Yeng Sung

Authors
  1. Susanne C. Brenner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. José C. Garay
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Li-Yeng Sung
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Susanne C. Brenner.

Ethics declarations

Conflict of interest

The authors have not disclosed any conflict of interests.

Additional information

Publisher's Note

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

This work was supported in part by the National Science Foundation under Grant No. DMS-19-13035.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Brenner, S.C., Garay, J.C. & Sung, LY. Multiscale Finite Element Methods for an Elliptic Optimal Control Problem with Rough Coefficients. J Sci Comput 91, 76 (2022). https://doi.org/10.1007/s10915-022-01834-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-022-01834-7

Keywords

Mathematics Subject Classification

Search

Navigation