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 setting. For , optimal convergence rates in the and norm are derived. Numerical experiments illustrate the theoretical error estimates and the applicability of the multiscale method to practical problems.
Keywords: Nonlinear monotone parabolic problem, multiple scales, heterogeneous multiscale method, finite elements, implicit Euler, fully discrete error, resonance error
@article{M2AN_2016__50_6_1659_0, author = {Abdulle, Assyr and Huber, Martin E.}, title = {Finite element heterogeneous multiscale method for nonlinear monotone parabolic homogenization problems}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {1659--1697}, publisher = {EDP-Sciences}, volume = {50}, number = {6}, year = {2016}, doi = {10.1051/m2an/2016003}, zbl = {1357.65169}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2016003/} }
TY - JOUR AU - Abdulle, Assyr AU - Huber, Martin E. TI - Finite element heterogeneous multiscale method for nonlinear monotone parabolic homogenization problems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 1659 EP - 1697 VL - 50 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2016003/ DO - 10.1051/m2an/2016003 LA - en ID - M2AN_2016__50_6_1659_0 ER -
%0 Journal Article %A Abdulle, Assyr %A Huber, Martin E. %T Finite element heterogeneous multiscale method for nonlinear monotone parabolic homogenization problems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 1659-1697 %V 50 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2016003/ %R 10.1051/m2an/2016003 %G en %F M2AN_2016__50_6_1659_0
Abdulle, Assyr; Huber, Martin E. Finite element heterogeneous multiscale method for nonlinear monotone parabolic homogenization problems. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 50 (2016) no. 6, pp. 1659-1697. doi : 10.1051/m2an/2016003. http://www.numdam.org/articles/10.1051/m2an/2016003/
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