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Journal of Scientific Computing

, Volume 60, Issue 3, pp 505–536 | Cite as

Coupled Model and Grid Adaptivity in Hierarchical Reduction of Elliptic Problems

  • Simona Perotto
  • Alessandro Veneziani
Article

Abstract

In this paper we propose a surrogate model for advection–diffusion–reaction problems characterized by a dominant direction in their dynamics. We resort to a hierarchical model reduction where we couple a modal representation of the transverse dynamics with a finite element approximation along the mainstream. This different treatment of the dynamics entails a surrogate model enhancing a purely 1D description related to the leading direction. The coefficients of the finite element expansion along this direction introduce a generally non-constant description of the transversal dynamics. Aim of this paper is to provide an automatic adaptive approach to locally select the dimension of the modal expansion as well as the finite element step in order to satisfy a prescribed tolerance on a goal functional of interest.

Keywords

Model reduction A posteriori modeling error analysis Mesh adaptivity Domain decomposition Finite elements 

Notes

Acknowledgments

The authors wish to thank warmly Alexandre Ern for many discussions and fundamental suggestions he gave along the entire preparation of the manuscript.

References

  1. 1.
    Achchab, B., Achchab, S., Agouzal, A.: Some remarks about the hierarchical a posteriori error estimate. Numer. Methods Partial Differ. Equ. 20(6), 919–932 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Ainsworth, M.: A posteriori error estimation for fully discrete hierarchic models of elliptic boundary value problems on thin domains. Numer. Math. 80, 325–362 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bangerth, W., Rannacher, R.: Adaptive Finite Element Methods for Differential Equations. Birkhauser Verlag, Basel (2003)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bank, R.E., Smith, R.K.: A posteriori error estimates based on hierarchical bases. SIAM J. Numer. Anal. 30, 921–935 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Becker, R., Rannacher, R.: An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10, 1–102 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Bernardi, C., Maday, Y., Patera, A.T.: Domain decomposition by the mortar element method. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 384. Kluwer, Dordrecht (1993)Google Scholar
  8. 8.
    Blanco, P.J., Leiva, J.S., Feijóo, R.A., Buscaglia, G.C.: Black-box decomposition approach for computational hemodynamics: one-dimensional models. Comput. Methods Appl. Mech. Eng. 200(13–16), 1389–1405 (2011)CrossRefzbMATHGoogle Scholar
  9. 9.
    Braack, M., Ern, A.: A posteriori control of modeling errors and discretization errors. Multiscale Model Simul. 1, 221–238 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Bruckstein, A.M., Donoho, D.L., Elad, M.: From sparse solutions of systems of equations to sparse modeling of signal and images. SIAM Rev. 51(1), 34–81 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Ciarlet, Ph: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)zbMATHGoogle Scholar
  12. 12.
    Clément, Ph: Approximation by finite element functions using local regularization. RAIRO Anal. Numer. 2, 77–84 (1975)Google Scholar
  13. 13.
    Cnossen, J.M., Bijl, H., van Brummelen, E.H.: Model-error estimation for goal-oriented model adaptation in flow-simulations. Finite volumes for complex applications IV, 173–183 (2005)Google Scholar
  14. 14.
    Dahmen, W., Kurdila, A.J., Oswald, P. (eds.): Multiscale Wavelet Methods for Partial Differential Equations. Wavelet Analysis and Its Applications, vol. 6. Academic Press, San Diego (1997)Google Scholar
  15. 15.
    Dörfler, W., Nochetto, R.H.: Small data oscillation implies the saturation assumption. Numer. Math. 91, 1–12 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Ern, A., Perotto, S., Veneziani, A.: Hierarchical model reduction for advection–diffusion–reaction problems. In: Kunisch, K., Of, G., Steinbach, O. (eds.) Numerical Mathematics and Advanced Applications, pp. 703–710. Springer, Berlin (2008)CrossRefGoogle Scholar
  17. 17.
    Formaggia, L., Nobile, F., Quarteroni, A., Veneziani, A.: Multiscale modelling of the circulatory system: a preliminary analysis. Comput. Visual. Sci. 2, 75–83 (1999)CrossRefzbMATHGoogle Scholar
  18. 18.
    Formaggia, L., Quarteroni, A., Veneziani, A. (eds.): Cardiovascular Mathematics: Modeling and Simulation of the Circulatory System. Modeling Simulation and Applications, vol. 1. Springer, Milano (2009)Google Scholar
  19. 19.
    Giles, M.B., Süli, E.: Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality. Acta Numer. 11, 145–236 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Hinze, M., Volkwein, S.: Proper orthogonal decomposition surrogate models for nonlinear dynamical systems: error estimates and suboptimal control. In: Dimension Reduction of Large-Scale Systems. Lect. Notes Comput. Sci. Eng., vol. 45, pp. 261–306. Springer, Berlin (2005)Google Scholar
  21. 21.
    Johnson, C.: A new paradigm for adaptive finite element methods. In: Whiteman, J. (eds.) Proceedings of MAFELAP 93. Wiley, New York (1993)Google Scholar
  22. 22.
    Lacour, C., Maday, Y.: Two different approaches for matching nonconforming grids: the mortar element method and the FETI method. BIT 37, 720–738 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Lasis, A., Süli, E.: Poincaré-type inequalities for broken Sobolev spaces. Tech. Report 03-10, Oxford University Computing Laboratory (2003)Google Scholar
  24. 24.
    Lions, J.L., Magenes, E.: Non Homogeneous Boundary Value Problems and Applications. Springer, Berlin (1972)CrossRefGoogle Scholar
  25. 25.
    Maday, Y., Ronquist, E.M.: A reduced-basis element method. C. R. Acad. Sci. Paris Ser. I(335), 195–200 (2002)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Micheletti, S., Perotto, S., David, F.: Model adaptation enriched with an anisotropic mesh spacing for nonlinear equations: application to environmental and CFD problems. Numer. Math. Theor. Meth. Appl. 6(3), 447–478 (2013)zbMATHMathSciNetGoogle Scholar
  27. 27.
    Miglio, E., Perotto, S., Saleri, F.: Model coupling techniques for free-surface flow problems. Part I. Nonlinear Anal. 63, 1885–1896 (2005)CrossRefGoogle Scholar
  28. 28.
    Oden, J.T., Prudhomme, S.: Goal-oriented error estimation and adaptivity for the finite element method. Comput. Math. Appl. 41, 735–756 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Oden, J.T., Prudhomme, S.: Estimation of modeling error in computational mechanics. J. Comput. Phys. 182, 469–515 (2002)CrossRefMathSciNetGoogle Scholar
  30. 30.
    Oden, J.T., Prudhomme, S., Hammerand, D.C., Kuczma, M.S.: Modeling error and adaptivity in nonlinear continuum mechanics. Comput. Methods Appl. Mech. Eng. 190, 6663–6684 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    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(1), 22–47 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Ohlberger, M., Smetana, K.: A new problem adapted hierarchical model reduction technique based on reduced basis methods and dimensional splitting. Tech. Report 03-10, University Muenster (2010)Google Scholar
  33. 33.
    Perotto, S.: Adaptive modeling for free-surface flows. M2AN Math. Model. Numer. Anal. 40(3), 469–499 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Perotto, S.: Hierarchical model (Hi-Mod) reduction in non-rectilinear domains. In press: Proceedings of the 21st International Conference on Domain Decomposition Methods. Springer, Berlin (2013)Google Scholar
  35. 35.
    Perotto, S., Ern, A., Veneziani, A.: Hierarchical local model reduction for elliptic problems: a domain decomposition approach. Multiscale Model. Simul. 8(4), 1102–1127 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Quarteroni, A., Valli, A.: Domain Decomposition Methods for Partial Differential Equations, Numerical Mathematics and Scientific Computation. Oxford University Press, New York (1999)Google Scholar
  37. 37.
    Robertson, A.M., Sequeira, A.: A director theory approach for modeling blood flow in the arterial system: an alternative to classical 1D models. Math. Models Methods Appl. Sci. 15(6), 871–906 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Stein, E., Rüter, M., Ohnimus, S.: Error-controlled adaptive goal-oriented modeling and finite element approximations in elasticity. Comput. Methods Appl. Mech. Eng. 196, 3598–3613 (2007)CrossRefzbMATHGoogle Scholar
  39. 39.
    Toselli, A., Widlund, O.: Domain Decomposition Methods–Algorithms and Theory. Springer, Berlin (2005)zbMATHGoogle Scholar
  40. 40.
    Vogelius, M., Babuška, I.: On a dimensional reduction method. III. A posteriori error estimation and an adaptive approach. Math. Comp. 37, 361–384 (1981)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.MOX, Dipartimento di Matematica “F. Brioschi”Politecnico di MilanoMilanItaly
  2. 2.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA

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