We study convergence of a mixed finite element–finite volume numerical scheme for the isentropic Navier–Stokes system under the full range of the adiabatic exponent. We establish suitable stability and consistency estimates and show that the Young measure generated by numerical solutions represents a dissipative measure-valued solutions of the limit system. In particular, using the recently established weak–strong uniqueness principle in the class of dissipative measure-valued solutions we show that the numerical solutions converge strongly to a strong solutions of the limit system as long as the latter exists.
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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. M. Lukáčová-Medvid’ová has been supported by the German Science Foundation under the grants LU 1470/2–3 and the Collaborative Research Centers TRR 146 and TRR 165.
Communicated by Eitan Tadmor.
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Feireisl, E., Lukáčová-Medvid’ová, M. Convergence of a Mixed Finite Element–Finite Volume Scheme for the Isentropic Navier–Stokes System via Dissipative Measure-Valued Solutions. Found Comput Math 18, 703–730 (2018). https://doi.org/10.1007/s10208-017-9351-2
- Compressible Navier–Stokes system
- Finite volume scheme
- Finite element scheme
- Measure-valued solution
Mathematics Subject Classification