Found Comput Math (2018) 18:703–730
Convergence of a Mixed Finite Element–Finite Volume
Scheme for the Isentropic Navier–Stokes System via
Dissipative Measure-Valued Solutions
· Mária Lukáˇcová-Medvid’ová
Received: 22 August 2016 / Revised: 20 February 2017 / Accepted: 3 April 2017 /
Published online: 19 April 2017
© SFoCM 2017
Abstract 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.
Keywords Compressible Navier–Stokes system · Finite volume scheme · Finite
element scheme · Stability · Convergence · Measure-valued solution
Mathematics Subject Classiﬁcation 65M12 · 65M60 · 76N10 · 35K61
Communicated by Eitan Tadmor.
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. Lukáˇcová-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.
Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25,
115 67 Praha 1, Czech Republic
Institute of Mathematics, Johannes Gutenberg-University Mainz, Staudingerweg 9, 55128 Mainz,