Selected topics in transport phenomena: deterministic and probabilistic aspects

Convergence of a mixed finite element-finite volume scheme for the compressible Navier-Stokes system via dissipative measure-valued solutions

Maria Lukacova

Universität Mainz
[SLIDES]

In this talk we discuss our recent results on the convergence of a mixed finite element-finite volume numerical scheme for the isentropic Navier-Stokes system under the full range of the adiabatic pressure 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. The present result is the first convergence result for numerical solutions of three-dimensional compressible isentropic Navier-Stokes equations in the case of full adiabatic exponent, i.e. also for the case when the existence of weak solutions is still open. If time permits we present the related error estimates for the case when the adiabatic exponent is larger than 3/2. Then it can be shown that the mixed finite element-finite volume scheme is asymptotic preserving in the singular limit when the Mach number approaches 0. The proof is based on the use of relative entropy. The present research has been obtained in the collaboration with E. Feireisl (Prague).