Abstract
In this paper, we propose a discretization for the nonsteady compressible Stokes Problem. This scheme is based on Crouzeix-Raviart approximation spaces. The discretization of the momentum balance is obtained by the usual finite element technique. The discrete mass balance is obtained by a finite volume scheme, with an upwinding of the density. The time discretization will be implicit in time. We prove the existence of a discrete solution. We prove that our scheme satisfies a discrete version of the relative entropy. As a consequence, we obtain an error estimate for this system. This preliminary work will be used in order to obtain a error estimate for the compressible Navier-Stokes system and has to the author’s knowledge not been studied previously.
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References
Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. Handb. Numer. Anal 7, 713–1018 (2000)
Feireisl, E., Jin, B.J., Novotny, A.: Relative entropies, suitable weak solutions, and weak strong uniqueness for the compressible Navier-Stokes system. http://arxiv.org/abs/1111.3082 (2011)
Feireisl, E., Novotny, A.: Weak-strong uniqueness property for the full Navier-Stokes-Fourier system. http://arxiv.org/abs/1111.4256 (2011)
Fettah, A., Gallouët, T.: Numerical approximation of the general compressible Stokes problem. IMA J. Numer. Anal. (2012)
Gallouët, T., Gastaldo, L., Herbin, R., Latché, J.C.: An unconditionally stable pressure correction scheme for the compressible barotropic Navier-Stokes equations. ESAIM. Math. Model. Numer. Anal. 42(02), 303–331 (2008)
Gallouët, T., Herbin, R., Latché, J.C.: A convergent finite element-finite volume scheme for the compressible Stokes problem part I : the isothermal case. http://arxiv.org/abs/0712.2798 (2007)
Herbin, R., Latché, J.C., Nguyen, T.T., et al.: Consistent explicit staggered schemes for compressible flows part I: the barotropic Euler equations. In: ESAIM Proceedings, Juillet 2013, vol. 40, pp. 83–102 (2013)
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Gallouët, T., Maltese, D., Novotný, A. (2014). Discrete Relative Entropy for the Compressible Stokes System. In: Fuhrmann, J., Ohlberger, M., Rohde, C. (eds) Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects. Springer Proceedings in Mathematics & Statistics, vol 77. Springer, Cham. https://doi.org/10.1007/978-3-319-05684-5_37
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DOI: https://doi.org/10.1007/978-3-319-05684-5_37
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