Homogenization of a pseudo-parabolic system via a spatial-temporal decoupling: Upscaling and corrector estimates for perforated domains

• Main Authors: Arthur. J. Vromans, Fons van de Ven, Adrian Muntean
• Format: Article
• Language: English
• Published: AIMS Press 2019-07-01
• Series: Mathematics in Engineering
• Subjects: periodic homogenization; pseudo-parabolic system; mixture theory; upscaled system; corrector estimates; perforated domains
• Online Access: https://www.aimspress.com/article/10.3934/mine.2019.3.548/fulltext.html

We determine corrector estimates quantifying the convergence speed of the upscaling of a pseudo-parabolic system containing drift terms incorporating the separation of length scales with relative size $\epsilon\ll1$. To achieve this goal, we exploit a natural spatial-temporal decomposition, which splits the pseudo-parabolic system into an elliptic partial differential equation and an ordinary differential equation coupled together. We obtain upscaled model equations, explicit formulas for effective transport coefficients, as well as corrector estimates delimitating the quality of the upscaling. Finally, for special cases we show convergence speeds for global times, i.e., $t\in\mathbf{R}_+$, by using time intervals expanding to the whole $\mathbf{R}_+$ simultaneously with passing to the homogenization limit $\epsilon\downarrow0$.