# Applied Mathematics Colloquium: Dr Ivan Yotov

## University of Pittsburgh

Friday, April 1, 2022

2:00 PM – 3:00 PM

2:00 PM – 3:00 PM

Title: Stokes-Biot modeling of fluid-poroelastic structure interaction

Abstract: We study mathematical models and their finite element approximations

for solving the coupled problem arising in the interaction between a

free fluid and a fluid in a poroelastic material. Applications of

interest include flows in fractured poroelastic media, coupling of

surface and subsurface flows, and arterial flows. The free fluid flow

is governed by the Navier-Stokes or Stokes/Brinkman equations, while

the poroelastic material is modeled using the Biot system of

poroelasticity. The two regions are coupled via dynamic and kinematic

interface conditions, including balance of forces, continuity of

normal velocity, and no-slip or slip with friction tangential velocity

condition. Well posedness of the weak formulations is established

using techniques from semigroup theory for evolution PDEs with

monotone operators. Mixed finite element methods are employed for the

numerical approximation. Solvability, stability, and accuracy of the

methods are analyzed with the use of suitable discrete inf-sup

conditions. Numerical results will be presented to illustrate the

performance of the methods, including their flexibility and robustness

for several applications of interest.

Abstract: We study mathematical models and their finite element approximations

for solving the coupled problem arising in the interaction between a

free fluid and a fluid in a poroelastic material. Applications of

interest include flows in fractured poroelastic media, coupling of

surface and subsurface flows, and arterial flows. The free fluid flow

is governed by the Navier-Stokes or Stokes/Brinkman equations, while

the poroelastic material is modeled using the Biot system of

poroelasticity. The two regions are coupled via dynamic and kinematic

interface conditions, including balance of forces, continuity of

normal velocity, and no-slip or slip with friction tangential velocity

condition. Well posedness of the weak formulations is established

using techniques from semigroup theory for evolution PDEs with

monotone operators. Mixed finite element methods are employed for the

numerical approximation. Solvability, stability, and accuracy of the

methods are analyzed with the use of suitable discrete inf-sup

conditions. Numerical results will be presented to illustrate the

performance of the methods, including their flexibility and robustness

for several applications of interest.

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