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

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.

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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)

    MathSciNet  Article  MATH  Google 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 modeling

  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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

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

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Allaire, G., Piatnitski, A.: Homogenization of the schrödinger equation and effective mass theorems. Commun. Math. Phys. 258(1), 1–22 (2005)

    Article  MATH  Google 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)

    Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Beavers, G.S., Joseph, D.D.: Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30(1), 197–207 (1967)

    Article  Google 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)

    Google 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)

    MathSciNet  Article  MATH  Google 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)

    Google 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)

  15. 15.

    Bove, R., Ubertini, S.: Modeling solid oxide fuel cell operation: approaches, techniques and results. J. Power Sources 159(1), 543–559 (2006)

    Article  Google Scholar 

  16. 16.

    Brenner, S.C., Scott, L.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, 2nd edn. Springer, Berlin (2002)

  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)

    MathSciNet  Article  MATH  Google 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)

    Article  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Ciarlet, P.G.: Finite Element Method for Elliptic Problems. Society for Industrial and Applied Mathematics, Philadelphia (2002)

    Google 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)

  23. 23.

    Cioranescu, D., Donato, P., Zaki, R.: The periodic unfolding method in perforated domains. Portugaliae Mathematica 63(4), 467–496 (2006)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Cook, B.: Introduction to fuel cells and hydrogen technology. Eng. Sci. Educ. J. 11, 205–216 (2002)

    Article  Google 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)

    Article  Google Scholar 

  26. 26.

    Goll, C.: Design of numerical methods for simulation models of a solid oxide fuel cell. Ph.D. thesis, Heidelberg University (2015)

  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)

  28. 28.

    Hornung, U. (ed.): Homogenization and Porous Media. Springer-Verlag, New York, NY (1997)

    Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    Maier, M.: Duality-based adaptivity in heterogeneous multiscale finite element discretization. Ph.D. thesis, Heidelberg University (2015)

  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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    Google 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)

    MathSciNet  Article  MATH  Google 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)

  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)

    Article  MATH  Google 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)

  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)

    MathSciNet  Article  MATH  Google Scholar 

  42. 42.

    Richter, T., Wick, T.: Variational localizations of the dual weighted residual estimator. J. Comput. Appl. Math. 279, 192–208 (2015)

    MathSciNet  Article  MATH  Google 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)

    Article  MATH  Google Scholar 

  44. 44.

    Saffman, P.G.: On the boundary condition at the interface of a porous medium. Stud. Appl. Math. 1, 93–101 (1971)

    Article  MATH  Google Scholar 

  45. 45.

    Schäfer, M., Turek, J.: Benchmark Computations of Laminar Flow Around a Cylinder. Vieweg, Braunschweig (1996)

    Google Scholar 

  46. 46.

    Tartar, L.: The general theory of homogenization. A personalized introduction. Springer, Berlin (2009)

    Google 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)

    Article  Google Scholar 

  48. 48.

    Zhikov, V., Kozlov, S., Oleinik, O.: Homogenization of Differential Operators and Integral Functionals. Springer, Berlin (1994)

    Google Scholar 

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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).

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Correspondence to Thomas Carraro.

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Carraro, T., Goll, C. A Goal-Oriented Error Estimator for a Class of Homogenization Problems. J Sci Comput 71, 1169–1196 (2017). https://doi.org/10.1007/s10915-016-0338-y

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Keywords

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