Finite element heterogeneous multiscale method for nonlinear monotone parabolic homogenization problems

Abdulle, Assyr; Huber, Martin E.

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 6, pp. 1659-1697.

We propose a multiscale method based on a finite element heterogeneous multiscale method (in space) and the implicit Euler integrator (in time) to solve nonlinear monotone parabolic problems with multiple scales due to spatial heterogeneities varying rapidly at a microscopic scale. The multiscale method approximates the homogenized solution at computational cost independent of the small scale by performing numerical upscaling (coupling of macro and micro finite element methods). Taking into account the error due to time discretization as well as macro and micro spatial discretizations, the convergence of the method is proved in the general $L^p\left(W^{1,p}\right)$ setting. For $p=2$, optimal convergence rates in the $L^2\left(H^1\right)$ and $C^0\left(L^2\right)$ norm are derived. Numerical experiments illustrate the theoretical error estimates and the applicability of the multiscale method to practical problems.

Classification : 65N30, 65M60, 74Q10, 74D10
Mots clés : Nonlinear monotone parabolic problem, multiple scales, heterogeneous multiscale method, finite elements, implicit Euler, fully discrete error, resonance error

A. Abdulle, On a priori error analysis of fully discrete heterogeneous multiscale FEM. Multiscale Model. Simul. 4 (2005) 447–459. | DOI | Zbl

A. Abdulle, The finite element heterogeneous multiscale method: a computational strategy for multiscale PDEs, in Multiple scales problems in biomathematics, mechanics, physics and numerics. Vol. 31 of GAKUTO Internat. Ser. Math. Sci. Appl. Gakkōtosho, Tokyo (2009) 133–181. | Zbl

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 | Zbl

A. Abdulle, W.E.B. Engquist and E. Vanden-Eijnden, The heterogeneous multiscale method. Acta Numer. 21 (2012) 1–87. | DOI | Zbl

A. Abdulle, M.J. Grote and C. Stohrer, Finite element heterogeneous multiscale method for the wave equation: long-time effects. Multiscale Model. Simul. 12 (2014) 1230–1257. | DOI | Zbl

A. Abdulle and M.E. Huber, Error estimates for finite element approximations of nonlinear monotone elliptic problems with application to numerical homogenization. Methods Partial Differ. Equ. 32 (2016) 955–969. | DOI | Zbl

A. Abdulle and A. Nonnenmacher, A short and versatile finite element multiscale code for homogenization problems. Comput. Methods Appl. Mech. Engrg. 198 (2009) 2839–2859. | DOI | Zbl

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. | DOI | Zbl

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 | Zbl

A. Abdulle, M.E. Huber and G. Vilmart, Linearized numerical homogenization method for nonlinear monotone parabolic multiscale problems. Multiscale Model. Simul. 13 (2015) 916–952. | DOI | Zbl

J.W. Barrett and W.B. Liu, Finite element approximation of the parabolic $p$-Laplacian. SIAM J. Numer. Anal. 31 (1994) 413–428. | DOI | Zbl

J.W. Barrett and W.B. Liu, Quasi-norm error bounds for the finite element approximation of a non-Newtonian flow. Numer. Math. 68 (1994) 437–456. | DOI | Zbl

A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic analysis for periodic structures. North-Holland Publishing Co., Amsterdam (1978). | Zbl

S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods. Vol. 15 of Texts Appl. Math., 3rd edn. Springer, New York (2008). | Zbl

V. Chiadò Piat, G. Dal Maso and A. Defranceschi, $G$-convergence of monotone operators. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 7 (1990) 123–160. | DOI | Numdam | Zbl

P.G. Ciarlet, The finite element method for elliptic problems. Vol. 4 of Stud. Math. Appl. North-Holland (1978). | Zbl

P.G. Ciarlet and P.A. Raviart, The combined effect of curved boundaries and numerical integration in isoparametric finite element methods, in The mathematical foundations of the finite element method with applications to partial differential equations (1972) 409–474. | Zbl

G. Dal Maso and A. Defranceschi, Correctors for the homogenization of monotone operators. Differ. Integral Equ. 3 (1990) 1151–1166. | Zbl

J.E. Dendy Jr., Galerkin’s method for some highly nonlinear problems. SIAM J. Numer. Anal. 14 (1977) 327–347. | DOI | Zbl

L. Diening, C. Ebmeyer and M. Růžička, Optimal convergence for the implicit space-time discretization of parabolic systems with $p$-structure. SIAM J. Numer. Anal. 45 (2007) 457–472. | DOI | Zbl

M. Dobrowolski, $L^{\infty }$-convergence of linear finite element approximation to nonlinear parabolic problems. SIAM J. Numer. Anal. 17 (1980) 663–674. | DOI | Zbl

R. Du and P. Ming, Heterogeneous multiscale finite element method with novel numerical integration schemes. Commun. Math. Sci. 8 (2010) 863–885. | DOI | Zbl

W. E and B. Engquist, The heterogeneous multiscale methods. Commun. Math. Sci. 1 (2003) 87–132. | DOI | Zbl

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. | Zbl

Y. Efendiev and A. Pankov, Numerical homogenization and correctors for nonlinear elliptic equations. SIAM J. Appl. Math. 65 (2004) 43–68. | DOI | Zbl

Y. Efendiev and A. Pankov, Numerical homogenization of nonlinear random parabolic operators. Multiscale Model. Simul. 2 (2004) 237–268. | DOI | Zbl

Y. Efendiev, T.Y. Hou and V. Ginting, Multiscale finite element methods for nonlinear problems and their applications. Commun. Math. Sci. 2 (2004) 553–589. | DOI | Zbl

J. Frehse and R. Rannacher, Asymptotic $L^{\infty }$-error estimates for linear finite element approximations of quasilinear boundary value problems. SIAM J. Numer. Anal. 15 (1978) 418–431. | DOI | Zbl

A. Gloria, An analytical framework for the numerical homogenization of monotone elliptic operators and quasiconvex energies. Multiscale Model. Simul. 5 (2006) 996–1043. | DOI | Zbl

A. Gloria, Reduction of the resonance error. Part 1: Approximation of homogenized coefficients. Math. Models Methods Appl. Sci. 21 (2011) 1601–1630. | DOI | Zbl

P. Henning and M. Ohlberger, A Newton-scheme framework for multiscale methods for nonlinear elliptic homogenization problems, in Proc. of the ALGORITMY 2012, 19th Conference on Scientific Computing. Edited by Vysoké Tatry, Podbanské (2012) 65–74. | Zbl

P. Henning and M. Ohlberger, Error control and adaptivity for heterogeneous multiscale approximations of nonlinear monotone problems. Discrete Contin. Dyn. Syst. Ser. S 8 (2015) 119–150. | Zbl

V.H. Hoang, Sparse finite element method for periodic multiscale nonlinear monotone problems. Multiscale Model. Simul. 7 (2008) 1042–1072. | DOI | Zbl

V.H. Hoang and C. Schwab, High-dimensional finite elements for elliptic problems with multiple scales. Multiscale Model. Simul. 3 (2005) 168–194. | DOI | Zbl

P. Houston, J. Robson and E. Süli, Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems. I. The scalar case. IMA J. Numer. Anal. 25 (2005) 726–749. | DOI | Zbl

M.E. Huber, Numerical homogenization methods for advection-diffusion and nonlinear monotone problems with multiple scales. Ph.D. thesis, École Polytechnique Fédérale de Lausanne (2015).

V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of differential operators and integral functionals. Springer-Verlag, Berlin, Heidelberg (1994). | Zbl

O.A. Ladyzhenskaya and N.N. Ural$\text{'}$tseva, Linear and quasilinear elliptic equations. Translated from the Russian by Scripta Technica. Inc. Translation editor: Leon Ehrenpreis. Academic Press, New York-London (1968). | Zbl

P. Ming and P. Zhang, Analysis of the heterogeneous multiscale method for parabolic homogenization problems. Math. Comp. 76 (2007) 153–177. | DOI | Zbl

I. Niyonzima, R.V. Sabariego, P. Dular and C. Geuzaine, Finite element computational homogenization of nonlinear multiscale materials in magnetostatics. IEEE Trans. Magn. 48 (2012) 587–590. | DOI

I. Niyonzima, R.V. Sabariego, P. Dular, F. Henrotte and C. Geuzaine, Computational homogenization for laminated ferromagnetic cores in magnetodynamics. IEEE Trans. Magn. 49 (2013) 2049–2052. | DOI

A. Pankov, $G$-convergence and homogenization of nonlinear partial differential operators. Vol. 422 of Math. Appl. Kluwer Academic Publishers, Dordrecht (1997). | Zbl

P.A. Raviart, The use of numerical integration in finite element methods for solving parabolic equations, in Topics in numerical analysis. Proc. of the Royal Irish Academy, Conference on Numerical Analysis (1972). Edited by J.J.H. Miller. Academic Press (1973) 233–264. | Zbl

N. Svanstedt, $G$-convergence of parabolic operators. Nonlinear Anal. 36 (1999) 807–842. | DOI | Zbl

N. Svanstedt, N. Wellander and J. Wyller, A numerical algorithm for nonlinear parabolic equations with highly oscillating coefficients. Numer. Methods Partial Differ. Equ. 12 (1996) 423–440. | DOI | Zbl

L. Tartar, Cours Peccot. Collège de France (1977).

L. Tartar, The general theory of homogenization. A personalized introduction. Vol. 7 of Lect. Notes Unione Matematica Italiana. Springer-Verlag, Berlin, UMI, Bologna (2009). | Zbl

V. Thomée, Galerkin finite element methods for parabolic problems. Vol. 25 of Springer Ser. Comput. Math. 2nd edn. Springer-Verlag, Berlin (2006).

M.F. Wheeler, A priori $L^2$ error estimates for Galerkin approximations to parabolic partial differential equations. SIAM J. Numer. Anal. 10 (1973) 723–759. | DOI | Zbl

J. Xu, Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal. 33 (1996) 1759–1777. | DOI | Zbl

X. Yue and W. E, The local microscale problem in the multiscale modeling of strongly heterogeneous media: effects of boundary conditions and cell size. J. Comput. Phys. 222 (2007) 556–572. | DOI | Zbl

E. Zeidler, Nonlinear functional analysis and its applications. II/A. Linear monotone operators, Translated from the German by the author and Leo F. Boron. Springer-Verlag, New York (1990). | Zbl

E. Zeidler, Nonlinear functional analysis and its applications. II/B. Nonlinear monotone operators, Translated from the German by the author and Leo F. Boron. Springer-Verlag, New York (1990). | Zbl