# American Institute of Mathematical Sciences

February  2015, 8(1): 91-118. doi: 10.3934/dcdss.2015.8.91

## Reduced basis finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems

 1 ANMC, Section de Mathématiques, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland, Switzerland 2 École Normale Supérieure de Rennes, INRIA Rennes, IRMAR, CNRS, UEB, Campus de Ker Lann, 35170 Bruz, France

Received  April 2013 Revised  November 2013 Published  July 2014

The reduced basis finite element heterogeneous multiscale method (RB-FE-HMM) for a class of nonlinear homogenization elliptic problems of nonmonotone type is introduced. In this approach, the solutions of the micro problems needed to estimate the macroscopic data of the homogenized problem are selected by a greedy algorithm and computed in an offline stage. It is shown that the use of reduced basis (RB) for nonlinear numerical homogenization reduces considerably the computational cost of the finite element heterogeneous multiscale method (FE-HMM). As the precomputed microscopic functions depend nonlinearly on the macroscopic solution, we introduce a new a posteriori error estimator for the greedy algorithm that guarantees the convergence of the online Newton method. A priori error estimates and uniqueness of the numerical solution are also established. Numerical experiments illustrate the efficiency of the proposed method.
Citation: Assyr Abdulle, Yun Bai, Gilles Vilmart. Reduced basis finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems. Discrete and Continuous Dynamical Systems - S, 2015, 8 (1) : 91-118. doi: 10.3934/dcdss.2015.8.91
##### References:
 [1] A. Abdulle, On a priori error analysis of fully discrete heterogeneous multiscale FEM, SIAM, Multiscale Model. Simul., 4 (2005), 447-459. doi: 10.1137/040607137. [2] A. Abdulle, A priori and a posteriori error analysis for numerical homogenization: A unified framework, Ser. Contemp. Appl. Math. CAM, 16 (2011), 280-305. doi: 10.1142/9789814366892_0009. [3] A. Abdulle and Y. Bai, Reduced basis finite element heterogeneous multiscale method for high-order discretizations of elliptic homogenization problems, J. Comput. Phys., 231 (2012), 7014-7036. doi: 10.1016/j.jcp.2012.02.019. [4] A. Abdulle and Y. Bai, Adaptive reduced basis finite element heterogeneous multiscale method, Comput. Methods Appl. Mech. Engrg., 257 (2013), 203-220. doi: 10.1016/j.cma.2013.01.002. [5] A. Abdulle, Y. Bai and G. Vilmart, An offline-online homogenization strategy to solve quasilinear two-scale problems at the cost of one-scale problems, Int. J. Numer. Meth. Engng., 99 (2014), 469-546. doi: 10.1002/nme.4682. [6] A. Abdulle and A. Nonnenmacher, A short and versatile finite element multiscale code for homogenization problem, Comput. Methods Appl. Mech. Engrg., 198 (2009), 2839-2859. doi: 10.1016/j.cma.2009.03.019. [7] A. Abdulle and C. Schwab, Heterogeneous multiscale FEM for diffusion problems on rough surfaces, SIAM, Multiscale Model. Simul., 3 (2005), 195-220. doi: 10.1137/030600771. [8] A. Abdulle and G. Vilmart, The effect of numerical integration in the finite element method for nonmonotone nonlinear elliptic problems with application to numerical homogenization methods, C. R. Acad. Sci. Paris, Ser. I, 349 (2011), 1041-1046. doi: 10.1016/j.crma.2011.09.005. [9] A. Abdulle and G. Vilmart, Coupling heterogeneous multiscale FEM with Runge-Kutta methods for parabolic homogenization problems: A fully discrete space-time analysis, Math. Models Methods Appl. Sci., 22 (2012), 1250002, 40 pp. doi: 10.1142/S0218202512500029. [10] A. Abdulle and G. Vilmart, A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems, Numer. Math., 121 (2012), 397-431. doi: 10.1007/s00211-011-0438-4. [11] A. Abdulle and G. Vilmart, Analysis of the finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems, Math. Comp., 83 (2014), 513-536. doi: 10.1090/S0025-5718-2013-02758-5. [12] F. Albrecht, B. Haasdonk, S. Kaulmann and M. Ohlberger, The localized reduced basis multiscale method, Proc. of ALGORITMY, (2012), 393-403. [13] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in Function Spaces, Differential Operators and Nonlinear Analysis (Friedrichroda, 1992), Teubner-Texte Math., 133, Teubner, Stuttgart, 1993, 9-126. doi: 10.1007/978-3-663-11336-2_1. [14] M. Artola and G. Duvaut, Un résultat d'homogénéisation pour une classe de problèmes de diffusion non linéaires stationnaires, Ann. Fac. Sci. Toulouse Math., 4 (1982), 1-28. doi: 10.5802/afst.572. [15] M. Barrault, Y. Maday, N. Nguyen and A. Patera, An ‘empirical interpolation method': Application to efficient reduced-basis discretization of partial differential equations, C. R. Acad. Sci. Paris Ser. I, 339 (2004), 667-672. doi: 10.1016/j.crma.2004.08.006. [16] J. Bear and Y. Bachmat, Introduction to Modelling of Transport Phenomena in Porous Media, Kluwer Academic, Dordrecht, The Netherlands, 1990. doi: 10.1007/978-94-009-1926-6. [17] A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland Publishing Co., Amsterdam, 1978. [18] P. Binev, A. Cohen, W. Dahmen, R. Devore, G. Petrova and P. Wojtaszczyk, Convergence rates for greedy algorithms in reduced basis methods, SIAM J. Math. Anal., 43 (2011), 1457-1472. doi: 10.1137/100795772. [19] L. Boccardo and F. Murat, Homogénéisation de problèmes quasi-linéaires, Publ. IRMA, Lille, 3 (1981), 13-51. [20] S. Boyaval, Reduced-basis approach for homogenization beyond the periodic setting, Multiscale Model. Simul., 7 (2008), 466-494. doi: 10.1137/070688791. [21] A. Buffa, Y. Maday, A. T. Patera, C. Prud'homme and G. Turinici, A priori convergence of the greedy algorithm for the parametrized reduced basis, ESAIM: M2AN, 46 (2012), 595-603. doi: 10.1051/m2an/2011056. [22] Z. Chen, W. Deng and H. Ye, Upscaling of a class of nonlinear parabolic equations for the flow transport in heterogeneous porous media, Commun. Math. Sci., 3 (2005), 493-515. doi: 10.4310/CMS.2005.v3.n4.a2. [23] Z. Chen and T. Y. Savchuk, Analysis of the multiscale finite element method for nonlinear and random homogenization problems, SIAM J. Numer. Anal., 46 (2007/08), 260-279. doi: 10.1137/060654207. [24] M. Chipot, Elliptic Equations: An Introductory Course, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-9982-5. [25] P. Ciarlet and P. Raviart, The combined effect of curved boundaries and numerical integration in isoparametric finite element methods, Math. Foundation of the FEM with Applications to PDE, (1972), 409-474. [26] W. E and B. Engquist, The heterogeneous multiscale methods, Commun. Math. Sci., 1 (2003), 87-132. doi: 10.4310/CMS.2003.v1.n1.a8. [27] W. E, P. Ming and P. Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems, J. Amer. Math. Soc., 18 (2005), 121-156. doi: 10.1090/S0894-0347-04-00469-2. [28] N. Fusco and G. Moscariello, On the homogenization of quasilinear divergence structure operators, Ann. Mat. Pura Appl., 146 (1987), 1-13. doi: 10.1007/BF01762357. [29] J. J. Douglas and T. Dupont, A Galerkin method for a nonlinear Dirichlet problem, Math. Comp., 29 (1975), 689-696. doi: 10.1090/S0025-5718-1975-0431747-2. [30] V. Jikov, S. Kozlov and O. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, Heidelberg, 1994. doi: 10.1007/978-3-642-84659-5. [31] 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), 3122-3137. doi: 10.1016/j.cma.2008.02.011. [32] S. Kaulmann, M. Ohlberger and B. Haasdonk, A new local reduced basis discontinuous Galerkin approach for heterogeneous multiscale problems, C. R. Math. Acad. Sci. Paris, 349 (2011), 1233-1238. doi: 10.1016/j.crma.2011.10.024. [33] Y. Maday, A. Patera and G. Turinici, A priori convergence theory for reduced-basis approximations of single-parameter elliptic partial differential equations, J. Sci. Comput., 17 (2002), 437-446. doi: 10.1023/A:1015145924517. [34] N. C. Nguyen, A multiscale reduced-basis method for parametrized elliptic partial differential equations with multiple scales, J. Comp. Phys., 227 (2008), 9807-9822. doi: 10.1016/j.jcp.2008.07.025. [35] J. A. Nitsche, On $L_{\infty }$-convergence of finite element approximations to the solution of a nonlinear boundary value problem, in Topics in Numerical Analysis, III (Proc. Roy. Irish Acad. Conf., Trinity Coll., Dublin, 1976), Academic Press, London, 1977, 317-325. [36] A. T. Patera and G. Rozza, Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations, to appear in (tentative rubric) MIT Pappalardo Graduate Monographs in Mechanical Engineering, 2007. [37] C. Prud'homme, D. V. Rovas, K. Veroy, L. Machiels, Y. Maday, A. T. Patera and G. Turinici, Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bounds methods, J. Fluids Eng., 124 (2002), 70-80. doi: 10.1115/1.1448332. [38] G. Rozza, D. Huynh and A. T. Patera, Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations, Arch. Comput. Methods. Eng., 15 (2008), 229-275. doi: 10.1007/s11831-008-9019-9.

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##### References:
 [1] A. Abdulle, On a priori error analysis of fully discrete heterogeneous multiscale FEM, SIAM, Multiscale Model. Simul., 4 (2005), 447-459. doi: 10.1137/040607137. [2] A. Abdulle, A priori and a posteriori error analysis for numerical homogenization: A unified framework, Ser. Contemp. Appl. Math. CAM, 16 (2011), 280-305. doi: 10.1142/9789814366892_0009. [3] A. Abdulle and Y. Bai, Reduced basis finite element heterogeneous multiscale method for high-order discretizations of elliptic homogenization problems, J. Comput. Phys., 231 (2012), 7014-7036. doi: 10.1016/j.jcp.2012.02.019. [4] A. Abdulle and Y. Bai, Adaptive reduced basis finite element heterogeneous multiscale method, Comput. Methods Appl. Mech. Engrg., 257 (2013), 203-220. doi: 10.1016/j.cma.2013.01.002. [5] A. Abdulle, Y. Bai and G. Vilmart, An offline-online homogenization strategy to solve quasilinear two-scale problems at the cost of one-scale problems, Int. J. Numer. Meth. Engng., 99 (2014), 469-546. doi: 10.1002/nme.4682. [6] A. Abdulle and A. Nonnenmacher, A short and versatile finite element multiscale code for homogenization problem, Comput. Methods Appl. Mech. Engrg., 198 (2009), 2839-2859. doi: 10.1016/j.cma.2009.03.019. [7] A. Abdulle and C. Schwab, Heterogeneous multiscale FEM for diffusion problems on rough surfaces, SIAM, Multiscale Model. Simul., 3 (2005), 195-220. doi: 10.1137/030600771. [8] A. Abdulle and G. Vilmart, The effect of numerical integration in the finite element method for nonmonotone nonlinear elliptic problems with application to numerical homogenization methods, C. R. Acad. Sci. Paris, Ser. I, 349 (2011), 1041-1046. doi: 10.1016/j.crma.2011.09.005. [9] A. Abdulle and G. Vilmart, Coupling heterogeneous multiscale FEM with Runge-Kutta methods for parabolic homogenization problems: A fully discrete space-time analysis, Math. Models Methods Appl. Sci., 22 (2012), 1250002, 40 pp. doi: 10.1142/S0218202512500029. [10] A. Abdulle and G. Vilmart, A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems, Numer. Math., 121 (2012), 397-431. doi: 10.1007/s00211-011-0438-4. [11] A. Abdulle and G. Vilmart, Analysis of the finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems, Math. Comp., 83 (2014), 513-536. doi: 10.1090/S0025-5718-2013-02758-5. [12] F. Albrecht, B. Haasdonk, S. Kaulmann and M. Ohlberger, The localized reduced basis multiscale method, Proc. of ALGORITMY, (2012), 393-403. [13] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in Function Spaces, Differential Operators and Nonlinear Analysis (Friedrichroda, 1992), Teubner-Texte Math., 133, Teubner, Stuttgart, 1993, 9-126. doi: 10.1007/978-3-663-11336-2_1. [14] M. Artola and G. Duvaut, Un résultat d'homogénéisation pour une classe de problèmes de diffusion non linéaires stationnaires, Ann. Fac. Sci. Toulouse Math., 4 (1982), 1-28. doi: 10.5802/afst.572. [15] M. Barrault, Y. Maday, N. Nguyen and A. Patera, An ‘empirical interpolation method': Application to efficient reduced-basis discretization of partial differential equations, C. R. Acad. Sci. Paris Ser. I, 339 (2004), 667-672. doi: 10.1016/j.crma.2004.08.006. [16] J. Bear and Y. Bachmat, Introduction to Modelling of Transport Phenomena in Porous Media, Kluwer Academic, Dordrecht, The Netherlands, 1990. doi: 10.1007/978-94-009-1926-6. [17] A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland Publishing Co., Amsterdam, 1978. [18] P. Binev, A. Cohen, W. Dahmen, R. Devore, G. Petrova and P. Wojtaszczyk, Convergence rates for greedy algorithms in reduced basis methods, SIAM J. Math. Anal., 43 (2011), 1457-1472. doi: 10.1137/100795772. [19] L. Boccardo and F. Murat, Homogénéisation de problèmes quasi-linéaires, Publ. IRMA, Lille, 3 (1981), 13-51. [20] S. Boyaval, Reduced-basis approach for homogenization beyond the periodic setting, Multiscale Model. Simul., 7 (2008), 466-494. doi: 10.1137/070688791. [21] A. Buffa, Y. Maday, A. T. Patera, C. Prud'homme and G. Turinici, A priori convergence of the greedy algorithm for the parametrized reduced basis, ESAIM: M2AN, 46 (2012), 595-603. doi: 10.1051/m2an/2011056. [22] Z. Chen, W. Deng and H. Ye, Upscaling of a class of nonlinear parabolic equations for the flow transport in heterogeneous porous media, Commun. Math. Sci., 3 (2005), 493-515. doi: 10.4310/CMS.2005.v3.n4.a2. [23] Z. Chen and T. Y. Savchuk, Analysis of the multiscale finite element method for nonlinear and random homogenization problems, SIAM J. Numer. Anal., 46 (2007/08), 260-279. doi: 10.1137/060654207. [24] M. Chipot, Elliptic Equations: An Introductory Course, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-9982-5. [25] P. Ciarlet and P. Raviart, The combined effect of curved boundaries and numerical integration in isoparametric finite element methods, Math. Foundation of the FEM with Applications to PDE, (1972), 409-474. [26] W. E and B. Engquist, The heterogeneous multiscale methods, Commun. Math. Sci., 1 (2003), 87-132. doi: 10.4310/CMS.2003.v1.n1.a8. [27] W. E, P. Ming and P. Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems, J. Amer. Math. Soc., 18 (2005), 121-156. doi: 10.1090/S0894-0347-04-00469-2. [28] N. Fusco and G. Moscariello, On the homogenization of quasilinear divergence structure operators, Ann. Mat. Pura Appl., 146 (1987), 1-13. doi: 10.1007/BF01762357. [29] J. J. Douglas and T. Dupont, A Galerkin method for a nonlinear Dirichlet problem, Math. Comp., 29 (1975), 689-696. doi: 10.1090/S0025-5718-1975-0431747-2. [30] V. Jikov, S. Kozlov and O. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, Heidelberg, 1994. doi: 10.1007/978-3-642-84659-5. [31] 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), 3122-3137. doi: 10.1016/j.cma.2008.02.011. [32] S. Kaulmann, M. Ohlberger and B. Haasdonk, A new local reduced basis discontinuous Galerkin approach for heterogeneous multiscale problems, C. R. Math. Acad. Sci. Paris, 349 (2011), 1233-1238. doi: 10.1016/j.crma.2011.10.024. [33] Y. Maday, A. Patera and G. Turinici, A priori convergence theory for reduced-basis approximations of single-parameter elliptic partial differential equations, J. Sci. Comput., 17 (2002), 437-446. doi: 10.1023/A:1015145924517. [34] N. C. Nguyen, A multiscale reduced-basis method for parametrized elliptic partial differential equations with multiple scales, J. Comp. Phys., 227 (2008), 9807-9822. doi: 10.1016/j.jcp.2008.07.025. [35] J. A. Nitsche, On $L_{\infty }$-convergence of finite element approximations to the solution of a nonlinear boundary value problem, in Topics in Numerical Analysis, III (Proc. Roy. Irish Acad. Conf., Trinity Coll., Dublin, 1976), Academic Press, London, 1977, 317-325. [36] A. T. Patera and G. Rozza, Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations, to appear in (tentative rubric) MIT Pappalardo Graduate Monographs in Mechanical Engineering, 2007. [37] C. Prud'homme, D. V. Rovas, K. Veroy, L. Machiels, Y. Maday, A. T. Patera and G. Turinici, Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bounds methods, J. Fluids Eng., 124 (2002), 70-80. doi: 10.1115/1.1448332. [38] G. Rozza, D. Huynh and A. T. Patera, Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations, Arch. Comput. Methods. Eng., 15 (2008), 229-275. doi: 10.1007/s11831-008-9019-9.
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