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Well-posedness and inverse Robin estimate for a multiscale elliptic/parabolic system
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).ORCID iD: 0000-0002-1752-1211
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).ORCID iD: 0000-0002-1160-0007
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).ORCID iD: 0000-0002-2185-641x
2018 (English)In: Applicable Analysis, ISSN 0003-6811, E-ISSN 1563-504X, Vol. 97, no 1, p. 89-106Article in journal (Refereed) Published
Abstract [en]

We establish the well-posedness of a coupled micro–macro parabolic– elliptic system modeling the interplay between two pressures in a gas–liquid mixture close to equilibrium that is filling a porous media with distributed microstructures. Additionally, we prove a local stability estimate for the inverse micro–macro Robin problem, potentially useful in identifying quantitatively a micro–macro interfacial Robin transfer coefficient given microscopic measurements on accessible fixed interfaces. To tackle the solvability issue we use two-scale energy estimates and twoscale regularity/compactness arguments cast in the Schauder’s fixed point theorem. A number of auxiliary problems, regularity, and scaling arguments are used in ensuring the suitable Fréchet differentiability of the solution and the structure of the inverse stability estimate.

Place, publisher, year, edition, pages
Taylor & Francis, 2018. Vol. 97, no 1, p. 89-106
Keywords [en]
Upscaled porous media, two-scale PDE, inverse micro–macro Robin problem
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kau:diva-62809DOI: 10.1080/00036811.2017.1364366ISI: 000417831700007OAI: oai:DiVA.org:kau-62809DiVA, id: diva2:1136119
Available from: 2017-08-25 Created: 2017-08-25 Last updated: 2021-03-31Bibliographically approved
In thesis
1. Multiscale models and simulations for diffusion and interaction in heterogeneous domains
Open this publication in new window or tab >>Multiscale models and simulations for diffusion and interaction in heterogeneous domains
2021 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

We investigate multiscale and multiphysics models for evolution systems in heterogeneous domains, with a focus on multiscale diffusions. Although diffusion is often studied in terms of continuum observables, it is a consequence of the motion of individual particles. Incorporating interactions between constituents and geometry often runs into complications, since interactions typically act on multiple length scales. We address this issue by studying different types of multiscale models and by applying them to a variety of scenarios known for their inherent complexity.

Our contributions can be grouped in two parts. In the first part, we pose two-scale reaction-diffusion systems in domains with varying microstructures. We prove well-posedness and construct finite element schemes with desirable approximation properties that resolve the microscopic domain variations and support parallel execution. In the second part of the thesis, we investigate certain interacting particle systems and their links to families of partial differential equations. In this spirit, we analyze a model of interacting populations, admitting dual descriptions from a system of ordinary differential equations and a porous media-like equation. We construct a multiscale simulation to evaluate scenarios in population dynamics. Finally, we investigate non-equilibrium dynamics and phase transitions within an interacting particle system in an extension of the classical Ehrenfest model.

Our overall focus is two-fold. On the one hand, we increase the theoretical understanding of multiscale models by providing modeling, analysis and simulation of specific two-scale couplings. On the other hand, we design computational frameworks and tailored implementations to improve the application of multiscale modeling to complex scenarios and large-scale systems. In this way, our contributions aim to expand the capacity of mathematical modeling to numerically approximate the rich and complex physical world.

Abstract [en]

We investigate multiscale and multiphysics models for evolution systems in heterogeneous domains. Our contributions can be grouped in two parts. First, we pose two-scale reaction-diffusion systems in domains with varying microstructures. We prove well-posedness and construct convergent and efficient finite element schemes that resolve the microscopic domain variations. Second, we investigate certain interacting particle systems and their links to a family of partial differential equations. We analyze a model of interacting populations, admitting dual descriptions from a system of ordinary differential equations and a porous media-like equation. We also construct a multiscale simulation to evaluate scenarios in population dynamics. Finally, we investigate non-equilibrium dynamics and phase transitions within a particle system extending the classical Ehrenfest model.

Our focus is two-fold: we increase the theoretical understanding of certain two-scale couplings, while on the other hand, we develop computational multiscale frameworks for a variety of scenarios known for their inherent complexity.

Place, publisher, year, edition, pages
Karlstad: Karlstads universitet, 2021. p. 212
Series
Karlstad University Studies, ISSN 1403-8099 ; 2021:10
Keywords
multiscale modeling, finite element methods, interacting particle sytems, population dynamics, non-equilibrium dynamics
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-83568 (URN)978-91-7867-195-3 (ISBN)978-91-7867-205-9 (ISBN)
Public defence
2021-05-19, Via Zoom, 15:00 (English)
Opponent
Supervisors
Available from: 2021-04-28 Created: 2021-03-31 Last updated: 2021-05-11Bibliographically approved

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Lind, MartinMuntean, AdrianRichardson, Omar

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