Journal of Scientific Computing

, Volume 76, Issue 2, pp 1188–1215 | Cite as

A Concurrent Global–Local Numerical Method for Multiscale PDEs

  • Yufang Huang
  • Jianfeng Lu
  • Pingbing Ming


We present a new hybrid numerical method for multiscale partial differential equations, which simultaneously captures the global macroscopic information and resolves the local microscopic events over regions of relatively small size. The method couples concurrently the microscopic coefficients in the region of interest with the homogenized coefficients elsewhere. The cost of the method is comparable to the heterogeneous multiscale method, while being able to recover microscopic information of the solution. The convergence of the method is proved for problems with bounded and measurable coefficients, while the rate of convergence is established for problems with rapidly oscillating periodic or almost-periodic coefficients. Numerical results are reported to show the efficiency and accuracy of the proposed method.


Concurrent global–local method Arlequin method Multiscale PDE H-convergence 

Mathematics Subject Classification

65N12 65N30 



The work of Lu is supported in part by the National Science Foundation under grant DMS-1454939. The work of Ming was supported by the National Natural Science Foundation of China for Distinguished Young Scholars 11425106, and by the National Natural Science Foundation of China grant 91230203, and by the funds from Creative Research Groups of China through grant 11321061, and by the support of CAS National Center for Mathematics and Interdisciplinary Sciences.


  1. 1.
    Abdulle, A., Weinan, E., Engquist, B., Vanden-Eijnden, E.: The heterogeneous multiscale method. Acta Numer. 21, 1–87 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abdulle, A., Jecker, O.: An optimization based heterogeneous to homogeneous coupling method. Commun. Math. Sci. 13, 1639–1648 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Abdulle, A., Jecker, O., Shapeev, A.: An optimization based coupling method for multiscale problems. Multisc. Model. Simul. 14, 1377–1416 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. Academic Press, London (2003)zbMATHGoogle Scholar
  5. 5.
    Apoung Kamga, J.-B., Pironneau, O.: Numerical zoom for multiscale problems with an application to nuclear waste disposal. J. Comput. Phys. 224, 403–413 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Babuška, I., Lipton, R.: L\(^2\)-global to local projection: an approach to multiscale analysis. Math. Models Methods Appl. Sci. 21, 2211–2226 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Babuška, I., Lipton, R., Stuebner, M.: The penetraion function and its application to microscale problems. BIT Numer. Math. 48, 167–187 (2008)CrossRefzbMATHGoogle Scholar
  8. 8.
    Babuška, I., Melenk, J.M.: The partition of unity finite element method. Int. J. Numer. Methods Eng. 40, 727–758 (1997)CrossRefzbMATHGoogle Scholar
  9. 9.
    Babuška, I., Motamed, M., Tempone, R.: A stochastic multiscale mehod for elastodynamic wave equation arising from fivre composites. Comput. Methods Appl. Mech. Eng. 276, 190–211 (2014)CrossRefGoogle Scholar
  10. 10.
    Ben Dhia, H.: Problèmes mécaniques multi-échelles: la méthode Arlequin. C. R. Acad. Sci. Paris Série II b 326, 899–904 (1998)zbMATHGoogle Scholar
  11. 11.
    Ben Dhia, H., Rateau, H.: The Arlequin method as a flexible engineering design tools. Int. J. Numer. Methods Eng. 62, 1442–1462 (2005)CrossRefzbMATHGoogle Scholar
  12. 12.
    Benssousan, A., Lions, J.L., Papanicolaou, G.: Asymptotic Analysis of Periodic Structures. North-Holland, Amsterdam (1978)Google Scholar
  13. 13.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)zbMATHGoogle Scholar
  14. 14.
    Demlow, A., Guzmán, J., Schatz, A.H.: Local energy estimates for the finite element method on sharply varying grids. Math. Comput. 80, 1–9 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Math. Sci. 136, 521–573 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Discacciati, M., Gervasio, P., Quarteroni, A.: Interface control domain decomposition methods for heterogeneous problems. Int. J. Numer. Methods Fluids 76, 471–496 (2014)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Du, Q., Gunzburger, M.D.: A gradient method approach to optimization-based multidisciplinary simulations and nonoverlapping domain decomposition algorithms. SIAM J. Numer. Anal. 37, 1513–1541 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Weinan, E.: Principles of Multiscale Modeling. Cambridge University Press, Cambridge (2011)zbMATHGoogle Scholar
  19. 19.
    Weinan, E., Engquist, B.: The heterogeneous multiscale methods. Commun. Math. Sci. 1, 87–132 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Weinan, E., Engquist, B., Li, X., Ren, W., Vanden-Eijnden, E.: Heterogeneous multiscale methods: a review. Commun. Comput. Phys. 2, 367–450 (2007)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Weinan, E., Ming, P.B., Zhang, P.W.: Analysis of the heterogeneous multiscale method for elliptic homogenization problems. J. Am. Math. Soc. 18, 121–156 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Gervasio, P., Lions, J.L., Quarteroni, A.: Heterogeneous coupling by virtual control methods. Numer. Math. 90, 241–264 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Glowinski, R., He, J., Rappaz, J., Wagner, J.: A multi-domain method for solving numerically multi-scale elliptic problems. C. R. Math. Acad. Sci. Paris 338, 741–746 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Guzmán, J., Sánchez, M.A., Sarkis, M.: On the accuracy of finite element approximations to a class of interface problems. Math. Comput. 85, 2071–2098 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kenig, C., Lin, F.H., Shen, Z.W.: Convergence rates in L\(^2\) for elliptic homogenization problems. Arch. Ration. Mech. Anal. 203, 1009–1036 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kuberry, P., Lee, H.: A decoupling algorithm for fluid-structure interaction problems based on optimization. Comput. Methods Appl. Mech. Eng. 267, 594–605 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Li, R., Ming, P.B., Tang, F.Y.: An efficient high order heterogeneous multiscale method for elliptic problems. Multisc. Model. Simul. 10, 259–286 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Lozinski, A., Pironneau, O.: Numerical zoom for advection diffusion problems with localized multiscales. Numer. Methods Partial Differ. Equ. 27, 197–207 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Lu, J., Ming, P.B.: Convergence of a force-based hybrid method in three dimensions. Commun. Pure Appl. Math. 66, 83–108 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Lu, J., Ming, P.B.: Convergence of a force-based hybrid method with planar sharp interface. SIAM J. Numer. Anal. 52, 2005–2026 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Meyers, N.G.: An L\(^p\)-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17(3), 189–206 (1963)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Ming, P.B., Yue, X.Y.: Numerical methods for multiscale elliptic problems. J. Comput. Phys. 214, 421–445 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Murat, F., Tartar, L.: H-convergence. In: Cherkaev, A., Kohn, R. (eds.) Topics in the Mathematical Modeling of Composite Materials, pp. 21–43. Birkhäuser, Boston (1997)CrossRefGoogle Scholar
  35. 35.
    Nitsche, J.A.: Ein kriterium für die quasioptimalität ds Ritzschen verfahrens. Nume. Math. 11, 346–348 (1968)CrossRefzbMATHGoogle Scholar
  36. 36.
    Nitsche, J.A., Schatz, A.H.: Interior estimates for Ritz-Galerkin methods. Math. Comput. 28, 937–958 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Oden, J.T., Vemaganti, K.S.: Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials, I. Error estimates and adaptive algorithms. J. Comput. Phys. 164, 22–47 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Oden, J.T., Vemaganti, K.S.: Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials, II. A computational environment for adaptively modeling of heterogeneous elastic solids. Computer Methods Appl. Mech. Eng. 190, 6089–6124 (2001)CrossRefzbMATHGoogle Scholar
  39. 39.
    Schwatz, A.H.: Perturbations of forms and error estimates for the finite element method at a point, with an application to improved superconvergence error estimates for subspaces that are symmetric with respect to a point. SIAM J. Numer. Anal. 42, 2342–2365 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Shen, Z.W.: Convergence rates and Hölder estimates in almost-periodic homogenization of elliptic systems. Anal. PDE 8, 1565–1601 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Tartar, L.: Compensated compactness and applications to partial differential equations. In: Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, vol. IV, (Editor), Res. Notes in Math., Vol. 39, Pitman, San Francisco, Calif., pp. 136–212 (1979)Google Scholar
  42. 42.
    Tartar, L.: The General Theory of Homogenization: A Personal Introduction. Springer, Berlin (2009)zbMATHGoogle Scholar

Copyright information

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Authors and Affiliations

  1. 1.Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  2. 2.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingChina
  3. 3.Department of MathematicsDuke UniversityDurhamUSA
  4. 4.Department of PhysicsDuke UniversityDurhamUSA
  5. 5.Department of ChemistryDuke UniversityDurhamUSA
  6. 6.The State Key Laboratory of Scientific and Engineering ComputingAcademy of Mathematics and Systems Science, Chinese Academy of SciencesBeijingChina

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