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

Arthur. J. Vromans; Fons van de Ven; Adrian Muntean; Arthur. J. Vromans; Fons van de Ven; Adrian Muntean

doi:10.3934/mine.2019.3.548

Mathematics in Engineering

2019, Volume 1, Issue 3: 548-582. doi: 10.3934/mine.2019.3.548

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Research article

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

1.
Centre for Analysis, Scientific computing and Applications (CASA), Eindhoven University of Technology, Den Dolech 2, 5612AZ, Eindhoven, The Netherlands
2.
Department of Mathematics and Computer Science, Karlstad University, Universitetsgatan 2, 651 88, Karlstad, Sweden

Received: 14 January 2019 Accepted: 02 June 2019 Published: 12 July 2019

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., $t\in{\bf{R}}_+$, by using time intervals expanding to the whole ${\bf{R}}_+$ simultaneously with passing to the homogenization limit $\epsilon\downarrow0$.
- periodic homogenization,
- pseudo-parabolic system,
- mixture theory,
- upscaled system,
- corrector estimates,
- perforated domains
Citation: Arthur. J. Vromans, Fons van de Ven, Adrian Muntean. Homogenization of a pseudo-parabolic system via a spatial-temporal decoupling: Upscaling and corrector estimates for perforated domains[J]. Mathematics in Engineering, 2019, 1(3): 548-582. doi: 10.3934/mine.2019.3.548

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Abstract

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., $t\in{\bf{R}}_+$, by using time intervals expanding to the whole ${\bf{R}}_+$ simultaneously with passing to the homogenization limit $\epsilon\downarrow0$.

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