Finite element heterogeneous multiscale method for nonlinear monotone parabolic homogenization problems

ANMC, Mathematics Section, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland.
Assyr.Abdulle@epfl.ch; Martin.Huber@epfl.ch

Received: 9 February 2015
Revised: 20 November 2015
Accepted: 6 January 2016

Abstract

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(W^1,p) setting. For p = 2, optimal convergence rates in the L²(H¹) and C⁰(L²) norm are derived. Numerical experiments illustrate the theoretical error estimates and the applicability of the multiscale method to practical problems.