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
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.