Abstract
AbstractThe aim of this work is the numerical homogenization of a parabolic problem with several time and spatial scales using the heterogeneous multiscale method. We replace the actual cell problem with an alternate one, using Dirichlet boundary and initial values instead of periodic boundary and time conditions. Further, we give a detailed a priori error analysis of the fully discretized, i.e., in space and time for both the macroscopic and the cell problem, method. Numerical experiments illustrate the theoretical convergence rates.
Funder
Karlsruher Institut für Technologie (KIT)
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Computational Mathematics,Computer Networks and Communications,Software
Reference30 articles.
1. Abdulle, A.: The finite element heterogeneous multiscale method: a computational strategy for multiscale PDEs. In: Multiple Scales Problems in Biomathematics, Mechanics, Physics and Numerics, International Series. Mathematical Sciences and Applications, vol. 31, pp. 133–181. Gakkotosho, Tokyo (2009)
2. Abdulle, A.: Numerical homogenization methods for parabolic monotone problems. In: Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations, Lecture Notes in Computational Science and Engineering, vol. 114, pp. 1–38. Springer, Cham (2016)
3. The heterogeneous multiscale method: Abdulle, A., E, W., Engquist, B., Vanden-Eijnden, E.: Acta Numer 21, 1–87 (2012). https://doi.org/10.1017/S0962492912000025
4. Abdulle, A., Huber, M.E.: Finite element heterogeneous multiscale method for nonlinear monotone parabolic homogenization problems. ESAIM Math. Model. Numer. Anal. 50(6), 1659–1697 (2016). https://doi.org/10.1051/m2an/2016003
5. Abdulle, A., Huber, M.E., Vilmart, G.: Linearized numerical homogenization method for nonlinear monotone parabolic multiscale problems. Multiscale Model. Simul. 13(3), 916–952 (2015). https://doi.org/10.1137/140975504