On the robustness of multiscale hybrid-mixed methods
HTML articles powered by AMS MathViewer
- by Diego Paredes, Frédéric Valentin and Henrique M. Versieux PDF
- Math. Comp. 86 (2017), 525-548 Request permission
Abstract:
In this work we prove uniform convergence of the Multiscale Hybrid-Mixed (MHM for short) finite element method for second-order elliptic problems with rough periodic coefficients. The MHM method is shown to avoid resonance errors without adopting oversampling techniques. In particular, we establish that the discretization error for the primal variable in the broken $H^1$ and $L^2$ norms are $O(h + \varepsilon ^\delta )$ and $O(h^2 + h \varepsilon ^\delta )$, respectively, and for the dual variable it is $O(h + \varepsilon ^\delta )$ in the $H(\operatorname {div};\cdot )$ norm, where $0<\delta \leq 1/2$ (depending on regularity). Such results rely on sharpened asymptotic expansion error estimates for the elliptic models with prescribed Dirichlet, Neumann or mixed boundary conditions.References
- Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
- Grégoire Allaire and Micol Amar, Boundary layer tails in periodic homogenization, ESAIM Control Optim. Calc. Var. 4 (1999), 209–243 (English, with English and French summaries). MR 1696289, DOI 10.1051/cocv:1999110
- Grégoire Allaire and Robert Brizzi, A multiscale finite element method for numerical homogenization, Multiscale Model. Simul. 4 (2005), no. 3, 790–812. MR 2203941, DOI 10.1137/040611239
- Rodolfo Araya, Christopher Harder, Diego Paredes, and Frédéric Valentin, Multiscale hybrid-mixed method, SIAM J. Numer. Anal. 51 (2013), no. 6, 3505–3531. MR 3143841, DOI 10.1137/120888223
- Todd Arbogast, Analysis of a two-scale, locally conservative subgrid upscaling for elliptic problems, SIAM J. Numer. Anal. 42 (2004), no. 2, 576–598. MR 2084227, DOI 10.1137/S0036142902406636
- Todd Arbogast and Kirsten J. Boyd, Subgrid upscaling and mixed multiscale finite elements, SIAM J. Numer. Anal. 44 (2006), no. 3, 1150–1171. MR 2231859, DOI 10.1137/050631811
- Todd Arbogast and Hailong Xiao, A multiscale mortar mixed space based on homogenization for heterogeneous elliptic problems, SIAM J. Numer. Anal. 51 (2013), no. 1, 377–399. MR 3033015, DOI 10.1137/120874928
- Ivo Babuška, Solution of interface problems by homogenization. I, SIAM J. Math. Anal. 7 (1976), no. 5, 603–634. MR 509273, DOI 10.1137/0507048
- I. Babuška and J. E. Osborn, Generalized finite element methods: their performance and their relation to mixed methods, SIAM J. Numer. Anal. 20 (1983), no. 3, 510–536. MR 701094, DOI 10.1137/0720034
- Alain Bensoussan, Jacques-Louis Lions, and George Papanicolaou, Asymptotic analysis for periodic structures, Studies in Mathematics and its Applications, vol. 5, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 503330
- Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
- Zhiming Chen and Thomas Y. Hou, A mixed multiscale finite element method for elliptic problems with oscillating coefficients, Math. Comp. 72 (2003), no. 242, 541–576. MR 1954956, DOI 10.1090/S0025-5718-02-01441-2
- C.-C. Chu, I. G. Graham, and T.-Y. Hou, A new multiscale finite element method for high-contrast elliptic interface problems, Math. Comp. 79 (2010), no. 272, 1915–1955. MR 2684351, DOI 10.1090/S0025-5718-2010-02372-5
- Weinan E and Bjorn Engquist, The heterogeneous multiscale methods, Commun. Math. Sci. 1 (2003), no. 1, 87–132. MR 1979846
- Yalchin R. Efendiev, Thomas Y. Hou, and Xiao-Hui Wu, Convergence of a nonconforming multiscale finite element method, SIAM J. Numer. Anal. 37 (2000), no. 3, 888–910. MR 1740386, DOI 10.1137/S0036142997330329
- Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383, DOI 10.1007/978-3-642-61623-5
- Georges Griso, Error estimate and unfolding for periodic homogenization, Asymptot. Anal. 40 (2004), no. 3-4, 269–286. MR 2107633
- P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
- C. Harder, A. L. Madureira, and F. Valentin, A hybrid-mixed method for elasticity, to appear in M2AN (http://dx.doi.org/10.1051/m2an/2015046).
- Christopher Harder, Diego Paredes, and Frédéric Valentin, A family of multiscale hybrid-mixed finite element methods for the Darcy equation with rough coefficients, J. Comput. Phys. 245 (2013), 107–130. MR 3066201, DOI 10.1016/j.jcp.2013.03.019
- Christopher Harder, Diego Paredes, and Frédéric Valentin, On a multiscale hybrid-mixed method for advective-reactive dominated problems with heterogeneous coefficients, Multiscale Model. Simul. 13 (2015), no. 2, 491–518. MR 3336297, DOI 10.1137/130938499
- Thomas Y. Hou, Xiao-Hui Wu, and Zhiqiang Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comp. 68 (1999), no. 227, 913–943. MR 1642758, DOI 10.1090/S0025-5718-99-01077-7
- Thomas Y. Hou and Xiao-Hui Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys. 134 (1997), no. 1, 169–189. MR 1455261, DOI 10.1006/jcph.1997.5682
- Thomas Y. Hou, Xiao-Hui Wu, and Yu Zhang, Removing the cell resonance error in the multiscale finite element method via a Petrov-Galerkin formulation, Commun. Math. Sci. 2 (2004), no. 2, 185–205. MR 2119937
- L. J. Jiang, Y. R. Efendiev, and I. Mishev, Mixed multiscale finite element methods using approximate global information based on partial upscaling, Computational Geosciences 14 (2010), no. 2, 319–341.
- V. V. Jikov, S. M. Kozlov, and O. A. Oleĭnik, Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994. Translated from the Russian by G. A. Yosifian [G. A. Iosif′yan]. MR 1329546, DOI 10.1007/978-3-642-84659-5
- Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968. Translated from the Russian by Scripta Technica, Inc; Translation editor: Leon Ehrenpreis. MR 0244627
- Claude Le Bris, Frédéric Legoll, and Alexei Lozinski, MsFEM à la Crouzeix-Raviart for highly oscillatory elliptic problems, Chinese Ann. Math. Ser. B 34 (2013), no. 1, 113–138. MR 3011462, DOI 10.1007/s11401-012-0755-7
- Yan Yan Li and Michael Vogelius, Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients, Arch. Ration. Mech. Anal. 153 (2000), no. 2, 91–151. MR 1770682, DOI 10.1007/s002050000082
- Norman G. Meyers, An $L^{p}$e-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 17 (1963), 189–206. MR 159110
- Shari Moskow and Michael Vogelius, First-order corrections to the homogenised eigenvalues of a periodic composite medium. A convergence proof, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), no. 6, 1263–1299. MR 1489436, DOI 10.1017/S0308210500027050
- D. Onofrei and B. Vernescu, Error estimates for periodic homogenization with non-smooth coefficients, Asymptot. Anal. 54 (2007), no. 1-2, 103–123. MR 2356467
- P.-A. Raviart and J. M. Thomas, Primal hybrid finite element methods for $2$nd order elliptic equations, Math. Comp. 31 (1977), no. 138, 391–413. MR 431752, DOI 10.1090/S0025-5718-1977-0431752-8
- Giancarlo Sangalli, Capturing small scales in elliptic problems using a residual-free bubbles finite element method, Multiscale Model. Simul. 1 (2003), no. 3, 485–503. MR 2030161, DOI 10.1137/S1540345902411402
- Marcus Sarkis and Henrique Versieux, Convergence analysis for the numerical boundary corrector for elliptic equations with rapidly oscillating coefficients, SIAM J. Numer. Anal. 46 (2008), no. 2, 545–576. MR 2383203, DOI 10.1137/060654773
- Mary Fanett Wheeler, Guangri Xue, and Ivan Yotov, A multiscale mortar multipoint flux mixed finite element method, ESAIM Math. Model. Numer. Anal. 46 (2012), no. 4, 759–796. MR 2891469, DOI 10.1051/m2an/2011064
Additional Information
- Diego Paredes
- Affiliation: Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso - IMA/ PUCV, Chile
- Email: diego.paredes@ucv.cl
- Frédéric Valentin
- Affiliation: Department of Computational and Applied Mathematics, National Laboratory for Scientific Computing - LNCC, Av. Getúlio Vargas, 333, 25651-070 Petrópolis - RJ, Brazil
- Email: valentin@lncc.br
- Henrique M. Versieux
- Affiliation: Instituto de Matemática, Universidade Federal do Rio de Janeiro - UFRJ, Rio de Janeiro - RJ, Brazil
- Email: henrique@im.ufrj.br
- Received by editor(s): October 15, 2014
- Received by editor(s) in revised form: May 31, 2015, and August 21, 2015
- Published electronically: March 28, 2016
- Additional Notes: The first author was partially supported by CONICYT/Chile through FONDECYT project 11140699 and PCI-CNPq/Brazil.
The second author was funded by CNPq/Brazil and CAPES/Brazil. - © Copyright 2016 American Mathematical Society
- Journal: Math. Comp. 86 (2017), 525-548
- MSC (2010): Primary 35J15, 65N15, 65N30
- DOI: https://doi.org/10.1090/mcom/3108
- MathSciNet review: 3584539