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Interplay Between Diffusion Anisotropy and Mesh Skewness in Hybrid High-Order Schemes Read chapter Read first chapter

2020 | OriginalPaper | Chapter

A Cell-Centered Finite Volume Method for the Navier–Stokes/Biot Model

Authors : Sergio Caucao, Tongtong Li, Ivan Yotov

Published in: Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples

Publisher: Springer International Publishing

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Abstract

We develop a cell-centered finite volume method for the Navier–Stokes/Biot model, based on a fully mixed formulation with weakly symmetric stresses. The multipoint stress mixed finite element method is employed for the Navier–Stokes and elasticity equations, while the multipoint flux mixed finite element method is used for Darcy’s flow. These methods allow for local elimination of the fluid and poroelastic stresses, vorticity, and rotation, resulting in a positive definite finite volume scheme for the fluid and structure velocities and the Darcy pressure, coupled via Lagrange multipliers on the interface to impose the transmission conditions.

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previous chapter Note on the Convergence of a Finite Volume Scheme for a Second Order Hyperbolic Equation with a Time Delay in Any Space Dimension
next chapter Convergence Study of a DDFV Scheme for the Navier-Stokes Equations Arising in the Domain Decomposition Setting
Literature
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Ambartsumyan, I., Ervin, V.J., Nguyen, T., Yotov, I.: A nonlinear Stokes–Biot model for the interaction of a non-Newtonian fluid with poroelastic media. ESAIM Math. Model. Numer. Anal. 53, 1915–1955 (2019)MathSciNetCrossRef
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Ambartsumyan, I., Khattatov, E., Nordbotten, J., Yotov, I.: A multipoint stress mixed finite element method for elasticity on simplicial grids. SIAM J. Numer. Anal. (To appear)
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Ambartsumyan, I., Khattatov, E., Yotov, I., Zunino, P.: A Lagrange multiplier method for a Stokes–Biot fluid-poroelastic structure interaction model. Numer. Math. 140(2), 513–553 (2018)
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Brezzi, F., Douglas Jr., J., Marini, L.D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47(2), 217–235 (1985)
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Camaño, J., García, C., Oyarzúa, R.: Analysis of a conservative mixed-FEM for the stationary Navier–Stokes problem. Preprint 2018-25, CI\(^2\) MA, Universidad de Concepción, Chile (2018)
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Caucao, S., Li, T., Yotov, I.: A multipoint stress-flux mixed finite element method for the Stokes–Biot model. In: preparation
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Gatica, G., Márquez, A., Oyarzúa, R., Rebolledo, R.: Analysis of an augmented fully-mixed approach for the coupling of quasi-newtonian fluids and porous media. Comput. Methods Appl. Mech. Engrg. 270, 76–112 (2014)MathSciNetCrossRef
8.
Wheeler, M.F., Yotov, I.: A multipoint flux mixed finite element method. SIAM J. Numer. Anal. 44(5), 2082–2106 (2006)MathSciNetCrossRef
Metadata
Title
A Cell-Centered Finite Volume Method for the Navier–Stokes/Biot Model
Authors
Sergio Caucao
Tongtong Li
Ivan Yotov
Copyright Year
2020
Book
Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples

Print ISBN: 978-3-030-43650-6

Electronic ISBN: 978-3-030-43651-3

Copyright Year: 2020

DOI
https://doi.org/10.1007/978-3-030-43651-3_29

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