Unconditional convergence and error estimates for bounded numerical solutions of the barotropic Navier-Stokes system

In this paper we develop a numerical scheme for approximating a $d$-dimensional chemotaxis-Navier-Stokes system, $d=2,3$, modeling cellular swimming in incompressible fluids. This model describes the chemotaxis-fluid interaction in cases where the chemical signal is consumed with a rate proportional to the amount of organisms. We construct numerical approximations based on the Finite Element method and analyze optimal error estimates and convergence towards regular solutions. In order to construct the numerical scheme, we use a splitting technique to deal with the chemo-attraction term in the cell-density equation, leading to introduce a new variable given by the gradient of the chemical concentration. Having the equivalent model, we consider a fully discrete Finite Element approximation which is well-posed and mass-conservative. We obtain uniform estimates and analyze the convergence of the scheme. Finally, we present some numerical simulations to verify the good behavior of our scheme, as well as to check numerically the optimal error estimates proved in our theoretical analysis.

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On the convergence of a finite volume method for the Navier–Stokes–Fourier system

IMA Journal of Numerical Analysis ◽

10.1093/imanum/draa060 ◽

2020 ◽

Author(s):

Eduard Feireisl ◽

Mária Lukáčová-Medviďová ◽

Hana Mizerová ◽

Bangwei She

Keyword(s):

Classical Solution ◽

Finite Volume ◽

Numerical Solutions ◽

Approximate Solutions ◽

Bounded Sequence ◽

Unconditional Convergence ◽

Navier Stokes ◽

Heat Conducting ◽

Polygonal Mesh ◽

Uniformly Bounded

Abstract The goal of the paper is to study the convergence of finite volume approximations of the Navier–Stokes–Fourier system describing the motion of compressible, viscous and heat-conducting fluids. The numerical flux uses upwinding with an additional numerical diffusion of order $\mathcal O(h^{ \varepsilon +1})$, $0<\varepsilon <1$. The approximate solutions are piecewise constant functions with respect to the underlying polygonal mesh. We show that the numerical solutions converge strongly to the classical solution as long as the latter exists. On the other hand, any uniformly bounded sequence of numerical solutions converges unconditionally to the classical solution of the Navier–Stokes–Fourier system without assuming a priori its existence. A similar unconditional convergence result is obtained for a sequence of numerical solutions with uniformly bounded densities and temperatures if the bulk viscosity vanishes.

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Convergence of a finite volume scheme for the compressible Navier–Stokes system

ESAIM Mathematical Modelling and Numerical Analysis ◽

10.1051/m2an/2019043 ◽

2019 ◽

Vol 53 (6) ◽

pp. 1957-1979 ◽

Cited By ~ 5

Author(s):

Eduard Feireisl ◽

Mária Lukáčová-Medvid’ová ◽

Hana Mizerová ◽

Bangwei She

Keyword(s):

Finite Volume ◽

Numerical Experiments ◽

Stokes System ◽

Numerical Solutions ◽

Navier Stokes ◽

Finite Volume Scheme ◽

Volume Scheme ◽

Benchmark Tests ◽

Stability And Consistency ◽

Theoretical Results

We study convergence of a finite volume scheme for the compressible (barotropic) Navier–Stokes system. First we prove the energy stability and consistency of the scheme and show that the numerical solutions generate a dissipative measure-valued solution of the system. Then by the dissipative measure-valued-strong uniqueness principle, we conclude the convergence of the numerical solution to the strong solution as long as the latter exists. Numerical experiments for standard benchmark tests support our theoretical results.

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