Finite element heterogeneous multiscale method for nonlinear monotone parabolic homogenization problems

Abdulle, Assyr¹ ; Huber, Martin E.¹

¹ ANMC, Mathematics Section, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland.

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

Résumé

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.

Détail

DOI : 10.1051/m2an/2016003

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