Advertisement

Journal of Scientific Computing

, Volume 71, Issue 3, pp 1169–1196 | Cite as

A Goal-Oriented Error Estimator for a Class of Homogenization Problems

  • Thomas CarraroEmail author
  • Christian Goll
Article

Abstract

We present a goal-oriented a posteriori error estimator for finite element approximations of a class of homogenization problems. As a rule, homogenization problems are defined through the coupling of a macroscopic solution and the solution of auxiliary problems. In this work we assume that the homogenized problem is known and that it depends on a finite number of auxiliary problems. The accuracy in the goal functional depends therefore on the discretization error of the macroscopic and the auxiliary solutions. We show that it is possible to compute the error contributions of all solution components separately and use this information to balance the different discretization errors. Additionally, we steer a local mesh refinement for both the macroscopic problem and the auxiliary problems. The high efficiency of this approach is shown by numerical examples. These include the upscaling of a periodic diffusion tensor, the case of a Stokes flow over a porous bed, and the homogenization of a fuel cell model which includes the flow in a gas channel over a porous substrate coupled with a multispecies nonlinear transport equation.

Keywords

A posteriori error estimation Dual based adaptivity Finite element method Homogenization problems Stokes/Darcy coupling Fuel cell application 

Notes

Acknowledgements

TC was supported by the German Research Council (DFG) through project “Multiscale modeling and numerical simulations of Lithium ion battery electrodes using real microstructures” (CA 633/2-1).

References

  1. 1.
    Abdulle, A., Nonnenmacher, A.: A posteriori error analysis of the heterogeneous multiscale method for homogenization problems. Comptes Rendus Mathematique 347(17–18), 1081–1086 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abdulle, A., Nonnenmacher, A.: Adaptive finite element heterogeneous multiscale method for homogenization problems. Comput. Methods Appl. Mech. Eng. 200(37–40), 2710 – 2726 (2011). Special issue on modeling error estimation and adaptive modelingGoogle Scholar
  3. 3.
    Abdulle, A., Nonnenmacher, A.: A posteriori error estimates in quantities of interest for the finite element heterogeneous multiscale method. Numer. Methods Partial Differ. Equ. 29(5), 1629–1656 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Alexanderian, A., Rathinam, M., Rostamian, R.: Homogenization, symmetry, and periodization in diffusive random media. Acta Math. Sci. 32(1), 129–154 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23(6), 1482–1518 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Allaire, G., Piatnitski, A.: Homogenization of the schrödinger equation and effective mass theorems. Commun. Math. Phys. 258(1), 1–22 (2005)CrossRefzbMATHGoogle Scholar
  7. 7.
    Bangerth, W., Heister, T., Heltai, L., Kanschat, G., Kronbichler, M., Maier, M., Turcksin, B., Young, T.D.: The deal.ii library, version 8.1. arXiv preprint, arXiv:1312.2266 (2013)
  8. 8.
    Bangerth, W., Rannacher, R.: Adaptive Finite Element Methods for Differential Equations. Birkhäuser Verlag, Basel (2003)CrossRefzbMATHGoogle Scholar
  9. 9.
    Barth, W.L., Carey, G.F.: On a boundary condition for pressure-driven laminar flow of incompressible fluids. Int. J. Numer. Methods Fluids 54(11), 1313–1325 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Beavers, G.S., Joseph, D.D.: Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30(1), 197–207 (1967)CrossRefGoogle Scholar
  11. 11.
    Becker, R., Braack, M., Meidner, D., Rannacher, R., Vexler, B.: Adaptive finite element methods for PDE-constrained optimal control problems. In: Jäger, W., Rannacher, R., Warnatz, J. (eds.) Reactive Flows, Diffusion and Transport, pp. 177–205. Springer, Berlin (2007)CrossRefGoogle Scholar
  12. 12.
    Becker, R., Rannacher, R.: An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10, 1–102 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bensoussan, A., Lions, J.L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures. Studies in Mathematics and Its Applications. North-Holland Pub. Co., New York (1978)zbMATHGoogle Scholar
  14. 14.
    Bourgeat, A., Piatnitski, A.: Approximations of effective coefficients in stochastic homogenization. Annales de l’Institut Henri Poincare (B) Probability and Statistics 40(2), 153–165 (2004)Google Scholar
  15. 15.
    Bove, R., Ubertini, S.: Modeling solid oxide fuel cell operation: approaches, techniques and results. J. Power Sources 159(1), 543–559 (2006)CrossRefGoogle Scholar
  16. 16.
    Brenner, S.C., Scott, L.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, 2nd edn. Springer, Berlin (2002)Google Scholar
  17. 17.
    Carraro, T., Goll, C., Marciniak-Czochra, A.: Mikelic̀ A.: Pressure jump interface law for the Stokes-Darcy coupling: confirmation by direct numerical simulations. J. Fluid Mech. 732, 510–536 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Carraro, T., Heuveline, V., Rannacher, R.: Determination of kinetic parameters in laminar flow reactors. I. Theoretical aspects. In: Rannacher, W.J.R., Warnatz, J. (eds.) Reactive Flows. Diffusion and Transport. Springer, Berlin (2007)Google Scholar
  19. 19.
    Carraro, T., Joos, J., Rüger, B., Weber, A., Ivers-Tiffée, E.: 3D finite element model reconstructed mixed-conducting cathodes: I. Performance quantication. Electrochim. Acta 77, 315–323 (2012)CrossRefGoogle Scholar
  20. 20.
    Carstensen, C., Verfurth, R.: Edge residuals dominate a posteriori error estimates for low order finite element methods. SIAM J. Numer. Anal. 36(5), 1571–1587 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ciarlet, P.G.: Finite Element Method for Elliptic Problems. Society for Industrial and Applied Mathematics, Philadelphia (2002)CrossRefzbMATHGoogle Scholar
  22. 22.
    Cioranescu, D., Donato, P.: An introduction to Homogenization. No. 17 in Oxford Lecture Series in Mathematics and Its Applications ; 17 ; Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, Oxford [u.a.] (1999)Google Scholar
  23. 23.
    Cioranescu, D., Donato, P., Zaki, R.: The periodic unfolding method in perforated domains. Portugaliae Mathematica 63(4), 467–496 (2006)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Cook, B.: Introduction to fuel cells and hydrogen technology. Eng. Sci. Educ. J. 11, 205–216 (2002)CrossRefGoogle Scholar
  25. 25.
    Ender, M., Joos, J., Carraro, T., Ivers-Tiffée, E.: Quantitative characterization of LiFePO4 cathodes reconstructed by FIB/SEM tomography. J. Electrochem. Soc. 159(7), A972–A980 (2012)CrossRefGoogle Scholar
  26. 26.
    Goll, C.: Design of numerical methods for simulation models of a solid oxide fuel cell. Ph.D. thesis, Heidelberg University (2015)Google Scholar
  27. 27.
    Hirschfelder, J., Curtiss, C.: Theory of propagation of flames. Part I: General equations. Symp. Combust. Flame Explos. Phenom. 3(1), 121–127 (1948)Google Scholar
  28. 28.
    Hornung, U. (ed.): Homogenization and Porous Media. Springer-Verlag, New York, NY (1997)zbMATHGoogle Scholar
  29. 29.
    Jäger, W., Mikelić, A.: On the interface boundary condition of Beavers, Joseph, and Saffman. SIAM J. Appl. Math. 60(4), 1111–1127 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Jäger, W., Mikelić, A., Neuss, N.: Asymptotic analysis of the laminar viscous flow over a porous bed. SIAM J. Sci. Comput. 22(6), 2006–2028 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Maier, M.: Duality-based adaptivity in heterogeneous multiscale finite element discretization. Ph.D. thesis, Heidelberg University (2015)Google Scholar
  32. 32.
    Marciniak-Czochra, A., Mikelić, A.: Effective pressure interface law for transport phenomena between an unconfined fluid and a porous medium using homogenization. Multiscale Model. Simul. 10(2), 285–305 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Oden, J.T., Prudhomme, S., Romkes, A., Bauman, P.T.: Multiscale modeling of physical phenomena: adaptive control of models. SIAM J. Sci. Comput 28, 2359–2389 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Ohlberger, M.: A posteriori error estimates for the heterogeneous multiscale finite element method for elliptic homogenization problems. Multiscale Model. Simul. 4(1), 88–114 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Rannacher, R.: Adaptive finite element discretization of flow problems for goal-oriented model reduction. In: Choi, H., Choi, H., Yoo, J. (eds.) Computational Fluid Dynamics 2008, pp. 31–45. Springer, Berlin Heidelberg (2009)CrossRefGoogle Scholar
  36. 36.
    Rannacher, R., Suttmeier, F.T.: A posteriori error estimation and mesh adaptation for finite element models in elasto-plasticity. Comput. Methods Appl. Mech. Eng. 176(1–4), 333–361 (1999). doi: 10.1016/S0045-7825(98)00344-2. http://www.sciencedirect.com/science/article/pii/S0045782598003442
  37. 37.
    Rannacher, R., Vihharev, J.: Adaptive finite element analysis of nonlinear problems: balancing of discretization and iteration errors. J. Numer. Math. 21(1), 23–62 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Reitz, W.: Handbook of fuel cells: fundamentals, technology, and applications, (volume 2). In: Vielstich, W., Lamm, A., Gasteiger, H.A. (eds.). Mater. Manuf. Process. 22(6), 789–789 (2007)Google Scholar
  39. 39.
    Rheinboldt, W.C., Mesztenyi, C.K.: On a data structure for adaptive finite element mesh refinements. ACM Trans. Math. Softw. 6(2), 166–187 (1980)CrossRefzbMATHGoogle Scholar
  40. 40.
    Richter, T.: Parallel multigrid method for adaptive finite elements with application to 3D flow problems. Ph.D. thesis, Mathematisch-Naturwissenschaftliche Gesamtfakultät, Universität Heidelberg (2005)Google Scholar
  41. 41.
    Richter, T.: A posteriori error estimation and anisotropy detection with the dual-weighted residual method. Int. J. Numer. Methods Fluids 62(1), 90–118 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Richter, T., Wick, T.: Variational localizations of the dual weighted residual estimator. J. Comput. Appl. Math. 279, 192–208 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Sab, K., Nedjar, B.: Periodization of random media and representative volume element size for linear composites. Comptes Rendus Mécanique 333(2), 187–195 (2005)CrossRefzbMATHGoogle Scholar
  44. 44.
    Saffman, P.G.: On the boundary condition at the interface of a porous medium. Stud. Appl. Math. 1, 93–101 (1971)CrossRefzbMATHGoogle Scholar
  45. 45.
    Schäfer, M., Turek, J.: Benchmark Computations of Laminar Flow Around a Cylinder. Vieweg, Braunschweig (1996)CrossRefzbMATHGoogle Scholar
  46. 46.
    Tartar, L.: The general theory of homogenization. A personalized introduction. Springer, Berlin (2009)zbMATHGoogle Scholar
  47. 47.
    Tseronis, K., Kookos, I., Theodoropoulos, C.: Modelling mass transport in solid oxide fuel cell anodes: a case for a multidimensional dusty gas-based model. Chem. Eng. Sci. 63, 5626–5638 (2008)CrossRefGoogle Scholar
  48. 48.
    Zhikov, V., Kozlov, S., Oleinik, O.: Homogenization of Differential Operators and Integral Functionals. Springer, Berlin (1994)Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Institute for Applied MathematicsHeidelberg UniversityHeidelbergGermany

Personalised recommendations