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

TL;DR

This paper proves that a combined finite element and finite volume numerical scheme for the isentropic Navier-Stokes equations converges to the true solution by employing dissipative measure-valued solutions and the weak-strong uniqueness principle.

Contribution

It introduces a novel convergence analysis for a mixed finite element finite volume scheme using dissipative measure-valued solutions and the weak-strong uniqueness principle.

Findings

Numerical solutions generate dissipative measure-valued solutions.

Strong convergence to classical solutions is established under certain conditions.

The scheme is stable and consistent across the full range of the adiabatic exponent.

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.