Error estimates for a numerical method for the compressible Navier–Stokes system on sufficiently smooth domains

Feireisl, Eduard; Hošek, Radim; Maltese, David; Novotný, Antonín

doi:10.1051/m2an/2016022

Feireisl, Eduard ^1,² ; Hošek, Radim ^1,² ; Maltese, David ^1,² ; Novotný, Antonín ^1,²

¹ Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Praha 1, Czech Republic.
² Institut Mathématiques de Toulon, EA2134, University of Toulon, BP 20132, 839 57 La Garde, France.

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 1, pp. 279-319.

Résumé

We derive an a priori error estimate for the numerical solution obtained by time and space discretization by the finite volume/finite element method of the barotropic Navier–Stokes equations. The numerical solution on a convenient polyhedral domain approximating a sufficiently smooth bounded domain is compared with an exact solution of the barotropic Navier–Stokes equations with a bounded density. The result is unconditional in the sense that there are no assumed bounds on the numerical solution. It is obtained by the combination of discrete relative energy inequality derived in [T. Gallouët, R. Herbin, D. Maltese and A. Novotný, IMA J. Numer. Anal. 36 (2016) 543–592.] and several recent results in the theory of compressible Navier–Stokes equations concerning blow up criterion established in [Y. Sun, C. Wang and Z. Zhang, J. Math. Pures Appl. 95 (2011) 36–47] and weak strong uniqueness principle established in [E. Feireisl, B.J. Jin and A. Novotný, J. Math. Fluid Mech. 14 (2012) 717–730].

Zbl

DOI : 10.1051/m2an/2016022

Classification : 35Q30, 65N12, 65N30, 76N10, 76N15, 76M10, 76M12
Mots-clés : Navier–Stokes system, finite element numerical method, finite volume numerical method, error estimates

Affiliations des auteurs :

Feireisl, Eduard ^{1, 2} ; Hošek, Radim ^{1, 2} ; Maltese, David ^{1, 2} ; Novotný, Antonín ^{1, 2}

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     title = {Error estimates for a numerical method for the compressible {Navier{\textendash}Stokes} system on sufficiently smooth domains},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {279--319},
     publisher = {EDP-Sciences},
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Feireisl, Eduard; Hošek, Radim; Maltese, David; Novotný, Antonín. Error estimates for a numerical method for the compressible Navier–Stokes system on sufficiently smooth domains. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 1, pp. 279-319. doi : 10.1051/m2an/2016022. http://www.numdam.org/articles/10.1051/m2an/2016022/

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