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Convergence of finite volume schemes for compressible flows
via dissipative measure valued solutions

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Maria Lukacova (University of Mainz)

In this talk we discuss the convergence of suitable numerical schemes for both viscous and inviscid compressible flows.
A standard paradigm for the existence of solutions in fluid dynamics is based on the construction of sequences of approximate solutions or numerical schemes. However, if the underlying model does not provide enough information for the required regularity of the approximate sequence, we are facing the problem to show the scheme's convergence. In particular, for multidimensional problems fine scale oscillations persist, which prevents us to obtain compactness result. Consequently, the standard framework of integrable functions seems not to be appropriate in general. To overcome this problem we introduce the class of dissipative measure-valued solutions, which allows us to show the convergence of finite volume or combined finite volume-finite element schemes for multidimensional isentropic Euler and Navier-Stokes equations, respectively. On the other hand, using the weak-strong uniqueness result for the above systems we know, that the dissipative measure-valued solution coincides with the strong solution, if the latter exists. Consequently, our results show convergence of our numerical schemes to the strong solutions.
These results have been obtained in collaboration with Eduard Feireisl (Academy of Sciences, Prague) and supported by the Collaborative Research Center TRR 146 in Mainz.