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An implicit stabilized finite element method for the compressible Navier–Stokes equations using finite calculus

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Abstract

A new implicit stabilized formulation for the numerical solution of the compressible Navier–Stokes equations is presented. The method is based on the finite calculus (FIC) scheme using the Galerkin finite element method (FEM) on triangular grids. Via the FIC formulation, two stabilization terms, called streamline term and transverse term, are added to the original conservation equations in the space-time domain. The non-linear system of equations resulting from the spatial discretization is solved implicitly using a damped Newton method benefiting from the exact Jacobian matrix. The matrix system is solved at each iteration with a preconditioned GMRES method. The efficiency of the proposed stabilization technique is checked out in the solution of 2D inviscid and laminar viscous flow problems where appropriate solutions are obtained especially near the boundary layer and shock waves. The work presented here can be considered as a follow up of a previous work of the authors Kouhi, Oñate (Int J Numer Methods Fluids 74:872–897, 2014). In that paper, the stabilized Galerkin FEM based on the FIC formulation was derived for the Euler equations together with an explicit scheme. In the present paper, the extension of this work to the Navier–Stokes equations using an implicit scheme is presented.

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Acknowledgments

The first author would like to acknowledge the financial support provided by CIMNE. We express our gratitude to Drs. Roberto Flores and Enrique Ortega for helpful discussions and suggestions. This research was partially supported by the SAFECON project of the European Research Council.

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Correspondence to Mohammad Kouhi.

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Kouhi, M., Oñate, E. An implicit stabilized finite element method for the compressible Navier–Stokes equations using finite calculus. Comput Mech 56, 113–129 (2015). https://doi.org/10.1007/s00466-015-1161-2

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