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Original Article

A residual-free bubble formulation for nonlinear elliptic problems with oscillatory coefficients

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Pages 1461-1476 | Received 22 May 2017, Accepted 06 Aug 2018, Published online: 24 Aug 2018

References

  • A. Abdulle and G. Vilmart, A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems, Numer. Math. 121(3) (2012), pp. 397–431. doi:10.1007/s00211-011-0438-4. MR2929073.
  • G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal. 23(6) (1992), pp. 1482–1518. doi:10.1137/0523084. MR1185639 (93k:35022).
  • R. Araya, C. Harder, D. Paredes and F. Valentin, Multiscale hybrid-mixed method, SIAM J. Numer. Anal. 51(6) (2013), pp. 3505–3531. doi:10.1137/120888223. MR3143841.
  • M. Artola and G. Duvaut, Un résultat d'homogénéisation pour une classe de problèmes de diffusion non linéaires stationnaires, Ann. Fac. Sci. Toulouse Math. (5) 4(1) (1982), pp. 1–28. French, with English summary MR673637 (84j:35020). doi: 10.5802/afst.572
  • J.-L. Auriault and J. Lewandowska, Homogenization analysis of diffusion and adsorption macrotransport in porous media: macrotransport in the absence of advection, Géotechnique 43(3) (1993), pp. 457–469. doi: 10.1680/geot.1993.43.3.457
  • I. Babuška, G. Caloz and J.E. Osborn, Special finite element methods for a class of second order elliptic problems with rough coefficients, SIAM J. Numer. Anal. 31(4) (1994), pp. 945–981. doi:10.1137/0731051. MR1286212 (95g:65146).
  • I. Babuška and J.E. Osborn, Generalized finite element methods: their performance and their relation to mixed methods, SIAM J. Numer. Anal. 20(3) (1983), pp. 510–536. doi:10.1137/0720034. MR701094 (84h:65076).
  • C. Baiocchi, F. Brezzi and L.P. Franca, Virtual bubbles and Galerkin-least-squares type methods (Ga.L.S.), Comput. Methods Appl. Mech. Engrg. 105(1) (1993), pp. 125–141. doi:10.1016/0045-7825(93)90119-I. MR1222297 (94g:65058).
  • A. Bensoussan, J. Lions and G. Papanicolaou, Asymptotic analysis for periodic structures, Studies in Mathematics and its Applications, 5, North-Holland Publishing Co., Amsterdam (1978) MR503330 (82h:35001).
  • L. Boccardo and F. Murat, Remarques sur l'homogénéisation de certains problèmes quasi-linéaires, Portugal. Math. 41(1–4) (1982), pp. 535–562. (1984). French, with English summary, MR766874 (86a:35022).
  • S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods, 3rd ed., Vol. Texts in Applied Mathematics15, Springer, New York, 2008. doi:10.1007/978-0-387-75934-0. MR2373954 (2008m:65001).
  • H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. MR2759829 (2012a:35002).
  • F. Brezzi, Interacting with the subgrid world, Numerical Analysis 1999 (Dundee), Chapman & Hall/CRC Res. Notes Math., vol. 420, Chapman & Hall/CRC, Boca Raton, FL (2000), pp. 69–82, MR1751112 (2001b:65121).
  • F. Brezzi, M.O. Bristeau, L.P. Franca, M. Mallet and G. Rogé, A relationship between stabilized finite element methods and the Galerkin method with bubble functions, Comput. Methods Appl. Mech. Engrg. 96(1) (1992), pp. 117–129. doi:10.1016/0045-7825(92)90102-P. MR1159592 (92k:76056).
  • F. Brezzi, L.P. Franca, T.J.R. Hughes and A. Russo, b=∫g, Comput. Methods Appl. Mech. Engrg. 145(3–4) (1997), pp. 329–339. doi:10.1016/S0045-7825(96)01221-2. MR1456019.
  • F. Brezzi, T.J.R. Hughes, L.D. Marini, A. Russo and E. S”uli, A priori error analysis of residual-free bubbles for advection–diffusion problems, SIAM J. Numer. Anal. 36(6) (1999), pp. 1933–1948. doi:10.1137/S0036142998342367. MR1712145.
  • F. Brezzi and A. Russo, Choosing bubbles for advection-diffusion problems, Math. Models Methods Appl. Sci. 4(4) (1994), pp. 571–587. doi:10.1142/S0218202594000327. MR1291139 (95h:76079).
  • Z. Chen, Multiscale methods for elliptic homogenization problems, Numer. Methods Partial Differ. Eq. 22(2) (2006), pp. 317–360. doi:10.1002/num.20099. MR2201437 (2007b:65117).
  • Z. Chen and T.Y. Savchuk, Analysis of the multiscale finite element method for nonlinear and random homogenization problems, SIAM J. Numer. Anal. 46(1) (2007/08), pp. 260–279. doi:10.1137/060654207. MR2377263 (2008k:35020).
  • P.G. Ciarlet, Linear and nonlinear functional analysis with applications, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2013. MR3136903.
  • P.G. Ciarlet, Mathematical elasticity. Vol. II, Studies in Mathematics and its Applications, vol. 27, North-Holland Publishing Co., Amsterdam, 1997, Theory of plates. MR1477663 (99e:73001).
  • L. Díaz and R. Naulin, A proof of the Schauder–Tychonoff theorem, language=English, with English and Spanish summaries, Divulg. Mat. 14(1) (2006), pp. 47–57. MR2586360 (2010k:47118).
  • J. Douglas Jr. and T. Dupont, A Galerkin method for a nonlinear Dirichlet problem, Math. Comp. 29 (1975), pp. 689–696. MR0431747 (55 #4742). doi: 10.1090/S0025-5718-1975-0431747-2
  • Y. Efendiev and T.Y. Hou, Multiscale finite element methods, Surveys and Tutorials in the Applied Mathematical Sciences, Vol. 4, Springer, New York, 2009. Theory and Applications. MR2477579 (2010h:65224).
  • Y. Efendiev, T. Hou and V. Ginting, Multiscale finite element methods for nonlinear problems and their applications, Commun. Math. Sci. 2(4) (2004), pp. 553–589. MR2119929 (2005m:65265). doi: 10.4310/CMS.2004.v2.n4.a2
  • Y.R. Efendiev, T.Y. Hou and X. Wu, Convergence of a nonconforming multiscale finite element method, SIAM J. Numer. Anal. 37(3) (2000), pp. 888–910. doi:10.1137/S0036142997330329. MR1740386 (2002a:65176).
  • Y. Efendiev and A. Pankov, Numerical homogenization of monotone elliptic operators, Multiscale Model. Simul. 2(1) (2003), pp. 62–79. doi:10.1137/S1540345903421611. MR2044957 (2005a:65153).
  • L.C. Evans, Partial differential equations, 2nd ed., Vol. Graduate Studies in Mathematics 19, American Mathematical Society, Providence, RI, 2010. MR2597943 (2011c:35002).
  • C. Farhat, I. Harari and L.P. Franca, The discontinuous enrichment method, Comput. Methods Appl. Mech. Engrg. 190(48) (2001), pp. 6455–6479. doi:10.1016/S0045-7825(01)00232-8. MR1870426 (2002j:76083).
  • L.P. Franca, A.L. Madureira, L. Tobiska and F. Valentin, Convergence analysis of a multiscale finite element method for singularly perturbed problems, Multiscale Model. Simul. 4(3) (2005), pp. 839–866. doi:10.1137/040608490 (electronic). MR2203943 (2006k:65316).
  • L.P. Franca, A.L. Madureira and F. Valentin, Towards multiscale functions: enriching finite element spaces with local but not bubble-like functions, Comput. Methods Appl. Mech. Engrg. 194(27–29) (2005), pp. 3006–3021. doi:10.1016/j.cma.2004.07.029. MR2142535 (2006a:65159).
  • L.P. Franca and A. Russo, Deriving upwinding, mass lumping and selective reduced integration by residual-free bubbles, Appl. Math. Lett. 9(5) (1996), pp. 83–88. doi:10.1016/0893-9659(96)00078-X. MR1415477 (97e:65121).
  • D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR1814364.
  • C. Harder, D. Paredes and F. Valentin, A family of multiscale hybrid-mixed finite element methods for the Darcy equation with rough coefficients, J. Comput. Phys. 245 (2013), pp. 107–130. doi:10.1016/j.jcp.2013.03.019. MR3066201.
  • P. Henning, Convergence of MSFEM approximations for elliptic, non-periodic homogenization problems, Netw. Heterog. Media 7(3) (2012), pp. 503–524. doi:10.3934/nhm.2012.7.503. MR2982460.
  • T.Y. Hou and X. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys. 134(1) (1997), pp. 169–189. doi:10.1006/jcph.1997.5682. MR1455261 (98e:73132).
  • T.Y. Hou, X. Wu and Z. Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comp. 68(227) (1999), pp. 913–943. doi:10.1090/S0025-5718-99-01077-7. MR1642758 (99i:65126).
  • T.J.R. Hughes, G.R. Feijóo, L. Mazzei and J. Quincy, The variational multiscale method – a paradigm for computational mechanics, Comput. Methods Appl. Mech. Engrg. 166(1–2) (1998), pp. 3–24. doi:10.1016/S0045-7825(98)00079-6. MR1660141.
  • T.J.R. Hughes and G. Sangalli, Variational multiscale analysis: the fine-scale Green's function, projection, optimization, localization, and stabilized methods, SIAM J. Numer. Anal. 45(2) (2007), pp. 539–557. doi:10.1137/050645646. MR2300286.
  • A.L. Madureira, Abstract multiscale-hybrid-mixed methods, Calcolo 52(4) (2015), pp. 543–557. doi:10.1007/s10092-014-0129-5. MR3421669.
  • A.L. Madureira, A multiscale finite element method for partial differential equations posed in domains with rough boundaries, Math. Comp. 78(265) (2009), pp. 25–34. doi:10.1090/S0025-5718-08-02159-5. MR2448695 (2010a:65243).
  • A.L. Madureira, Numerical methods and analysis of multiscale problems, SpringerBriefs in Mathematics Springer, Cham, 2017. doi:10.1007/978-3-319-50866-5. MR3618733.
  • A. Masud and R. Calderer, A variational multiscale method for incompressible turbulent flows: bubble functions and fine scale fields, Comput. Methods Appl. Mech. Engrg. 200(33–36) (2011), pp. 2577–2593. doi:10.1016/j.cma.2011.04.010. MR2812026.
  • A. Masud and R.A. Khurram, A multiscale finite element method for the incompressible Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg. 195(13–16) (2006), pp. 1750–1777. doi:10.1016/j.cma.2005.05.048. MR2203991.
  • P. Ming and X. Yue, Numerical methods for multiscale elliptic problems, J. Comput. Phys. 214(1) (2006), pp. 421–445. doi:10.1016/j.jcp.2005.09.024. MR2208685 (2006j:65359).
  • J. Pousin and J. Rappaz, Consistency, stability, a priori and a posteriori errors for Petrov–Galerkin methods applied to nonlinear problems, Numer. Math. 69(2) (1994), pp. 213–231. doi:10.1007/s002110050088. MR1310318 (95k:65111).
  • J.V. de A. Ramalho, Novos métodos de elementos finitos enriquecidos aplicados a modelos de rea??o-advecção-difusão transientes, D.Sc. Thesis, Laboratório Nacional de Computação Científica, Petrópolis, RJ, 2005.
  • G. Sangalli, Capturing small scales in elliptic problems using a residual-free bubbles finite element method, Multiscale Model. Simul. 1(3) (2003), pp. 485–503. doi:10.1137/S1540345902411402 (electronic). MR2030161 (2004m:65202).
  • S. Tokarzewski and I. Andrianov, Effective coefficients for real non-linear and fictitious linear temperature-dependent periodic composites, Int. J. Non-Linear Mech. 36(1) (2001), pp. 187–195. doi:10.1016/S0020-7462(00)00012-3. MR1783661 (2001i:80011).
  • H. Versieux and M. Sarkis, A three-scale finite element method for elliptic equations with rapidly oscillating periodic coefficients, Domain decomposition methods in science and engineering XVI, Lect. Notes Comput. Sci. Eng., vol. 55, Springer, Berlin, 2007, pp. 763–770. doi:10.1007/978-3-540-34469-8_95. MR2334173.
  • H.M. Versieux and M. Sarkis, Numerical boundary corrector for elliptic equations with rapidly oscillating periodic coefficients, Comm. Numer. Methods Engrg. 22(6) (2006), pp. 577–589. doi: 10.1002/cnm.834. MR2235030 (2007d:65117).
  • W. E and B. Engquist, The heterogeneous multiscale methods, Commun. Math. Sci. 1(1) (2003), pp. 87–132. MR1979846 (2004b:35019). doi: 10.4310/CMS.2003.v1.n1.a8
  • W. E, P. Ming and P. Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems, J. Amer. Math. Soc. 18(1) (2005), pp. 121–156. doi:10.1090/S0894-0347-04-00469-2. MR2114818 (2005k:65246).
  • J. Xu, Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal. 33(5) (1996), pp. 1759–1777. doi:10.1137/S0036142992232949. MR1411848 (97i:65169).
  • T. Yu and X. Yue, Residual-free bubble methods for numerical homogenization of elliptic problems, Commun. Math. Sci. 9(4) (2011), pp. 1163–1176. doi:10.4310/CMS.2011.v9.n4.a12. MR2901822.

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