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

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

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

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].

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

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