Analysis of the finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems

Abdulle, Assyr; Vilmart, Gilles

doi:10.1090/S0025-5718-2013-02758-5

Analysis of the finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems

Authors: Assyr Abdulle and Gilles Vilmart
Journal: Math. Comp. 83 (2014), 513-536
MSC (2010): Primary 65N30, 65M60, 74D10, 74Q05
DOI: https://doi.org/10.1090/S0025-5718-2013-02758-5
Published electronically: August 30, 2013
MathSciNet review: 3143682
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: An analysis of the finite element heterogeneous multiscale method for a class of quasilinear elliptic homogenization problems of nonmonotone type is proposed. We obtain optimal convergence results for dimension $d\leq 3$. Our results, which also take into account the microscale discretization, are valid for both simplicial and quadrilateral finite elements. Optimal a priori error estimates are obtained for the $H^1$ and $L^2$ norms, error bounds similar to those for linear elliptic problems are derived for the resonance error. Uniqueness of a numerical solution is proved. Moreover, the Newton method used to compute the solution is shown to converge. Numerical experiments confirm the theoretical convergence rates and illustrate the behavior of the numerical method.

References

Assyr Abdulle, On a priori error analysis of fully discrete heterogeneous multiscale FEM, Multiscale Model. Simul. 4 (2005), no. 2, 447–459. MR 2162863, DOI https://doi.org/10.1137/040607137
Assyr Abdulle, Discontinuous Galerkin finite element heterogeneous multiscale method for elliptic problems with multiple scales, Math. Comp. 81 (2012), no. 278, 687–713. MR 2869033, DOI https://doi.org/10.1090/S0025-5718-2011-02527-5
A. Abdulle, The finite element heterogeneous multiscale method: a computational strategy for multiscale PDEs, Multiple scales problems in biomathematics, mechanics, physics and numerics, GAKUTO Internat. Ser. Math. Sci. Appl., vol. 31, Gakk\B{o}tosho, Tokyo, 2009, pp. 133–181. MR 2590959
Assyr Abdulle, A priori and a posteriori error analysis for numerical homogenization: a unified framework, Multiscale problems, Ser. Contemp. Appl. Math. CAM, vol. 16, Higher Ed. Press, Beijing, 2011, pp. 280–305. MR 2952740, DOI https://doi.org/10.1142/9789814366892_0009
Assyr Abdulle and Achim Nonnenmacher, A short and versatile finite element multiscale code for homogenization problems, Comput. Methods Appl. Mech. Engrg. 198 (2009), no. 37-40, 2839–2859. MR 2567848, DOI https://doi.org/10.1016/j.cma.2009.03.019
Assyr Abdulle and Christoph Schwab, Heterogeneous multiscale FEM for diffusion problems on rough surfaces, Multiscale Model. Simul. 3 (2004/05), no. 1, 195–220. MR 2123116, DOI https://doi.org/10.1137/030600771
Assyr Abdulle and Gilles Vilmart, A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems, Numer. Math. 121 (2012), no. 3, 397–431. MR 2929073, DOI https://doi.org/10.1007/s00211-011-0438-4
Assyr Abdulle, Weinan E, Björn Engquist, and Eric Vanden-Eijnden, The heterogeneous multiscale method, Acta Numer. 21 (2012), 1–87. MR 2916381, DOI https://doi.org/10.1017/S0962492912000025
N. André and M. Chipot, Uniqueness and nonuniqueness for the approximation of quasilinear elliptic equations, SIAM J. Numer. Anal. 33 (1996), no. 5, 1981–1994. MR 1411859, DOI https://doi.org/10.1137/S0036142994267400
Michel Artola and Georges Duvaut, Un résultat d’homogénéisation pour une classe de problèmes de diffusion non linéaires stationnaires, Ann. Fac. Sci. Toulouse Math. (5) 4 (1982), no. 1, 1–28 (French, with English summary). MR 673637

J. Bear and Y. Bachmat, Introduction to modelling of transport phenomena in porous media, Kluwer Academic, Dordrecht, The Netherlands, 1991.

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

L. Bers, F. John, and M. Schechter, Partial differential equations, Lectures in Applied Mathematics, Proceedings of the Summer Seminar, Boulder, CO, 1957.

L. Boccardo and F. Murat, Homogénéisation de problèmes quasi-linéaires, Publ. IRMA, Lille., 3(7) (1981), 13–51.

Alain Bourgeat and Andrey Piatnitski, Approximations of effective coefficients in stochastic homogenization, Ann. Inst. H. Poincaré Probab. Statist. 40 (2004), no. 2, 153–165 (English, with English and French summaries). MR 2044813, DOI https://doi.org/10.1016/S0246-0203%2803%2900065-7

Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR 2373954

Zhiming Chen, Weibing Deng, and Huang Ye, Upscaling of a class of nonlinear parabolic equations for the flow transport in heterogeneous porous media, Commun. Math. Sci. 3 (2005), no. 4, 493–515. MR 2188680

Zhangxin Chen and Tatyana Y. Savchuk, Analysis of the multiscale finite element method for nonlinear and random homogenization problems, SIAM J. Numer. Anal. 46 (2007/08), no. 1, 260–279. MR 2377263, DOI https://doi.org/10.1137/060654207

Michel Chipot, Elliptic equations: an introductory course, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2009. MR 2494977

P. G. Ciarlet, Basic error estimates for elliptic problems, Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991, pp. 17–351. MR 1115237

P. G. Ciarlet and P.-A. Raviart, The combined effect of curved boundaries and numerical integration in isoparametric finite element methods, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 409–474. MR 0421108

Jim Douglas Jr. and Todd Dupont, A Galerkin method for a nonlinear Dirichlet problem, Math. Comp. 29 (1975), 689–696. MR 431747, DOI https://doi.org/10.1090/S0025-5718-1975-0431747-2

Rui Du and Pingbing Ming, Heterogeneous multiscale finite element method with novel numerical integration schemes, Commun. Math. Sci. 8 (2010), no. 4, 863–885. MR 2744910

Weinan E and B. Engquist, The heterogeneous multi-scale methods, Commun. Math. Sci., 1 (2003), 87–132.

Weinan E, Pingbing Ming, and Pingwen Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems, J. Amer. Math. Soc. 18 (2005), no. 1, 121–156. MR 2114818, DOI https://doi.org/10.1090/S0894-0347-04-00469-2

Yalchin Efendiev and Thomas Y. Hou, Multiscale finite element methods, Surveys and Tutorials in the Applied Mathematical Sciences, vol. 4, Springer, New York, 2009. Theory and applications. MR 2477579

Y. Efendiev, T. Hou, and V. Ginting, Multiscale finite element methods for nonlinear problems and their applications, Commun. Math. Sci. 2 (2004), no. 4, 553–589. MR 2119929

M. Feistauer, M. Křížek, and V. Sobotíková, An analysis of finite element variational crimes for a nonlinear elliptic problem of a nonmonotone type, East-West J. Numer. Math. 1 (1993), no. 4, 267–285. MR 1318806

N. Fusco and G. Moscariello, On the homogenization of quasilinear divergence structure operators, Ann. Mat. Pura Appl. (4) 146 (1987), 1–13. MR 916685, DOI https://doi.org/10.1007/BF01762357

Antoine Gloria, Numerical approximation of effective coefficients in stochastic homogenization of discrete elliptic equations, ESAIM Math. Model. Numer. Anal. 46 (2012), no. 1, 1–38. MR 2846365, DOI https://doi.org/10.1051/m2an/2011018

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

A. Karageorghis and D. Lesnic, Steady-state nonlinear heat conduction in composite materials using the method of fundamental solutions, Comput. Methods Appl. Mech. Engrg. 197 (2008), no. 33-40, 3122–3137. MR 2427102, DOI https://doi.org/10.1016/j.cma.2008.02.011

O. A. Ladyzhenskaya, The boundary value problems of mathematical physics, Applied Mathematical Sciences, vol. 49, Springer-Verlag, New York, 1985. Translated from the Russian by Jack Lohwater [Arthur J. Lohwater]. MR 793735

J. A. Nitsche, On $L_{\infty }$-convergence of finite element approximations to the solution of a nonlinear boundary value problem, Topics in numerical analysis, III (Proc. Roy. Irish Acad. Conf., Trinity Coll., Dublin, 1976) Academic Press, London-New York, 1977, pp. 317–325. MR 513215

A.G. Whittington, A.M. Hofmeister, and P.I. Nabelek, Temperature-dependent thermal diffusivity of the Earth’s crust and implications for magmatism, Nature, 458 (2009), 319–321.

Jinchao Xu, Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal. 33 (1996), no. 5, 1759–1777. MR 1411848, DOI https://doi.org/10.1137/S0036142992232949

V. V. Yurinskiĭ, Averaging of symmetric diffusion in a random medium, Sibirsk. Mat. Zh. 27 (1986), no. 4, 167–180, 215 (Russian). MR 867870