### Abstract

Pseudo-parabolic equations have been used to model unsaturated

fluid flow in porous media. In this paper it is shown how a pseudo-parabolic

equation can be upscaled when using a spatio-temporal decomposition employed in the Peszy´nska-Showalter-Yi paper [8]. The spatial-temporal decomposition transforms the pseudo-parabolic equation into a system containing an elliptic partial differential equation and a temporal ordinary differential equation. To strengthen our argument, the pseudo-parabolic equation has been given advection/convection/drift terms. The upscaling is done with the technique of periodic homogenization via two-scale convergence. The wellposedness of the extended pseudo-parabolic equation is shown as well. Moreover, we argue that under certain conditions, a non-local-in-time term arises from the elimination of an unknown.

fluid flow in porous media. In this paper it is shown how a pseudo-parabolic

equation can be upscaled when using a spatio-temporal decomposition employed in the Peszy´nska-Showalter-Yi paper [8]. The spatial-temporal decomposition transforms the pseudo-parabolic equation into a system containing an elliptic partial differential equation and a temporal ordinary differential equation. To strengthen our argument, the pseudo-parabolic equation has been given advection/convection/drift terms. The upscaling is done with the technique of periodic homogenization via two-scale convergence. The wellposedness of the extended pseudo-parabolic equation is shown as well. Moreover, we argue that under certain conditions, a non-local-in-time term arises from the elimination of an unknown.

Translated title of the contribution | Periodieke homgoenizatie van een pseudoparabolische vergelijking via een spatiaal-temporele decompositie |
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Original language | English |

Title of host publication | Proceedings of The 20th European Conference on Mathematics for Industry |

Editors | I Farag ́o, F. Izs ́ak, P. Simon |

Publisher | Springer |

Number of pages | 6 |

Publication status | Published - 15 Apr 2019 |

Event | 20th European Conference on Mathematics for Industry - Budapest, Hungary Duration: 18 Jun 2018 → 22 Jun 2018 |

### Publication series

Name | The European Consortium for Mathematics in Industry Series |
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### Conference

Conference | 20th European Conference on Mathematics for Industry |
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Country | Hungary |

City | Budapest |

Period | 18/06/18 → 22/06/18 |

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## Cite this

Vromans, A., van de Ven, F., & Muntean, A. (2019). Periodic homogenization of a pseudo-parabolic equation via a spatial-temporal decomposition. In I. Farag ́o, F. Izs ́ak, & P. Simon (Eds.),

*Proceedings of The 20th European Conference on Mathematics for Industry*(The European Consortium for Mathematics in Industry Series). Springer.