Abstract
In this contribution, we formulate a heterogeneous multiscale finite element method (HMM) for monotone elliptic operators. This is done in the general concept of HMM, which was initially introduced by E and Engquist [E, Engquist, The heterogeneous multiscale methods, Commun. Math. Sci., 1(1):87–132, 2003]. Since the straightforward formulation is not suitable for a direct implementation in the nonlinear setting, we present a corresponding algorithm, which involves the computation of additional cell problems. The algorithm is validated by numerical experiments and can be used to effectively determine homogenized solutions.
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References
Abdulle, A.: The finite element heterogeneous multiscale method: a computational strategy for multiscale PDEs. In: Multiple Scales Problems in Biomathematics, Mechanics, Physics and Numerics. GAKUTO Internat. Ser. Math. Sci. Appl., Tokyo, vol. 31, pp. 133–181 (2009)
Abdulle, A., Engquist, B.: Finite element heterogeneous multiscale methods with near optimal computational complexity. Multiscale Model. Simul. 6(4), 1059–1084 (2007)
Bastian, P., Blatt, M., Dedner, A., Engwer, C., Klöfkorn, R., Kornhuber, R., Ohlberger, M., Sander, O.: A generic grid interface for parallel and adaptive scientific computing. Part II: Implementation and tests in DUNE. Computing 82, 121–138 (2008)
Dedner, A., Klöfkorn, R., Nolte, M., Ohlberger, M.: A generic interface for parallel and adaptive discretization schemes: abstraction principles and the Dune-Fem module. In: Computing, pp. 1–32. Springer Wien (2010)
Weinan, E., Engquist, B.: The heterogeneous multiscale methods. Commun. Math. Sci. 1(1), 87–132 (2003)
Efendiev, Y., Hou, T., Ginting, V.: Multiscale finite element methods for nonlinear problems and their applications. Commun. Math. Sci. 2(4), 553–589 (2004)
Henning, P., Ohlberger, M.: The heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains. Numer. Math. 113(4), 601–629 (2009)
Henning, P., Ohlberger, M.: A-posteriori error estimate for a heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift. Preprint Uni. Münster N09/09
Henning, P., Ohlberger, M.: The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift. Netw. Heterog. Media 5(4) (2010)
Ladyzhenskaya, O.A., Uraltseva, N.N.: Linear and quasilinear elliptic equations. Academic Press, New York (1968)
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)
Wall, P.: Some homogenization and corrector results for nonlinear monotone operators. J. Nonlinear Math. Phys. 5(3), 331–348 (1998)
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Henning, P., Ohlberger, M. (2012). On the Implementation of a Heterogeneous Multi-scale Finite Element Method for Nonlinear Elliptic Problems. In: Dedner, A., Flemisch, B., Klöfkorn, R. (eds) Advances in DUNE. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28589-9_11
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DOI: https://doi.org/10.1007/978-3-642-28589-9_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-28588-2
Online ISBN: 978-3-642-28589-9
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