# Journal of theKorean Mathematical SocietyJKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

## Article

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J. Korean Math. Soc. 2022; 59(3): 519-548

Published online May 1, 2022 https://doi.org/10.4134/JKMS.j210211

## Stabilized-penalized collocated finite volume scheme for incompressible biofluid flows

Nasserdine Kechkar, Mohammed Louaar

University Freres Mentouri; University of Science and Technology Houari Boumediene

### Abstract

In this paper, a stabilized-penalized collocated finite volume (SPCFV) scheme is developed and studied for the stationary generalized Navier-Stokes equations with mixed Dirichlet-traction boundary conditions modelling an incompressible biological fluid flow. This method is based on the lowest order approximation (piecewise constants) for both velocity and pressure unknowns. The stabilization-penalization is performed by adding discrete pressure terms to the approximate formulation. These simultaneously involve discrete jump pressures through the interior volume-boundaries and discrete pressures of volumes on the domain boundary. Stability, existence and uniqueness of discrete solutions are established. Moreover, a convergence analysis of the nonlinear solver is also provided. Numerical results from model tests are performed to demonstrate the stability, optimal convergence in the usual $L^2$ and discrete $H^1$ norms as well as robustness of the proposed scheme with respect to the choice of the given traction vector.

Keywords: Navier-Stokes equations, finite element, finite volume, stabilization

MSC numbers: 35Q35, 76D05, 65N08, 65N30

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