Volume 51, Number 1, January-February 2017
|Page(s)||279 - 319|
|Published online||15 December 2016|
Error estimates for a numerical method for the compressible Navier–Stokes system on sufficiently smooth domains∗,∗∗,∗∗∗
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.
Revised: 13 January 2016
Accepted: 23 March 2016
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].
Mathematics Subject Classification: 35Q30 / 65N12 / 65N30 / 76N10 / 76N15 / 76M10 / 76M12
Key words: Navier–Stokes system / finite element numerical method / finite volume numerical method / error estimates
The research of E. Feireisl leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC Grant Agreement 320078. The Institute of Mathematics of the Academy of Sciences of the Czech Republic is supported by RVO:67985840.
The research of R. Hošek leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC Grant Agreement 320078. The Institute of Mathematics of the Academy of Sciences of the Czech Republic is supported by RVO:67985840.
© EDP Sciences, SMAI 2016
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