E. Feireisl, M. Lukáčová-Medvid’ová, H. Mizerová, B. She:

Numerical analysis of compressible fluid flows (monograph)

Series: Modeling, Simulation, and Applications, Springer (2021)

The main message of our recently published book is the new approach in the numerical analysis of nonlinear equations arising in fluid dynamics in the spirit of the Lax Equivalence Principle in the context of linear problems: ”stability + consistency , convergence.”

It contains original results based on dissipative measure-valued solutions, weak-strong uniqueness and K−convergence which are used not only to prove convergence of various numerical schemes but also to solve the problem of oscillatory solutions.

E. Feireisl, M. Lukáčová-Medvid’ová, H. Mizerová:

Convergence of finite volume schemes for the Euler equations via dissipative measure–valued solutions,

Foundations of Computational Mathematics 20 (2020), p. 923–966;

DOI: https://doi.org/10.1007/s10208-019-09433-z

Since the multi-dimensional Euler equations are ill-posed in the class of weak solutions for L1−initial data, here we propose to investigate the convergence in the framework of dissipative measure–valued solutions. We study a class of entropy stable finite volume schemes for the barotropic and complete compressible Euler equations, and show that numerical solutions converge pointwise to the regular solution of the limit systems at least on the lifespan of the latter.

E. Feireisl, M. Lukáčová-Medvid’ová, H. Mizerová:

A finite volume scheme for the Euler system inspired by the two velocities approach,

Numerische Mathematik 144 (2020), p. 89-–132;

DOI: https://doi.org/10.1007/s00211-019-01078-y

As far as we know this is the first convergence result for a finite volume method for multi–dimensional Euler equations, where the convergence has been proven assuming only that the gas remains in its non–degenerate region.

E. Feireisl, M. Lukáčová-Medvid’ová, H. Mizerová, B. She:

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

ESAIM: Mathematical Modelling and Numerical Analysis 53(6) (2019), p. 1957–1979;

DOI: https://doi.org/10.1051/m2an/2019043

To the best of our knowledge this is the first proof of convergence of a finite volume method

for the multi-dimensional compressible barotropic Navier–Stokes equations assuming

only the existence of the strong solution. Moreover, the adiabatic coefficient stays in a

physically reasonable range between 1 and 2.

M. Lukácová-Medvid’ová, H. Mizerová, Š. Necasová, M. Renardy:

Global existence result for the generalized Peterlin viscoelastic model,

SIAM Journal on Mathematical Analysis 49(4) (2017), p. 2950–2964;

DOI: https://doi.org/10.1137/16M1068505

The paper contains a proof of existence of large-data global-in-time weak solutions to a class of differential models of viscoelastic flows with diffusive stress. Moreover, classical solutions are shown to exist under stronger assumptions in the case of creeping flows and two-dimensional flows.

E. Feireisl, M. Lukáčová-Medvid’ová, H. Mizerová, B. She:

Numerical analysis of compressible fluid flows (monograph)

Series: Modeling, Simulation, and Applications, Springer (2021)

E. Feireisl, M. Lukáčová-Medvid’ová, H. Mizerová:

Convergence of finite volume schemes for the Euler equations via dissipative measure–valued solutions,

Foundations of Computational Mathematics 20 (2020), p. 923–966;

DOI: https://doi.org/10.1007/s10208-019-09433-z

E. Feireisl, M. Lukáčová-Medvid’ová, H. Mizerová:

A finite volume scheme for the Euler system inspired by the two velocities approach,

Numerische Mathematik 144 (2020), p. 89-–132;

DOI: https://doi.org/10.1007/s00211-019-01078-y

E. Feireisl, M. Lukáčová-Medvid’ová, H. Mizerová, B. She:

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

ESAIM: Mathematical Modelling and Numerical Analysis 53(6) (2019), p. 1957–1979;

DOI: https://doi.org/10.1051/m2an/2019043

D. Basarić, E. Feireisl, M. Lukáčová-Medvid’ová, H. Mizerová, Y.Yuan:

Penalization method for the Navier-Stokes-Fourier system,

ESAIM: M2AN 56(6) (2022)

DOI: https://doi.org/10.1051/m2an/2022063

Beddrich, J., Süli, E., Wohlmuth, B.:

Numerical simulation of the time-fractional Fokker–Planck equation and applications to polymeric fluids,

Journal of Computational Physics, 2024, 497

DOI: 10.1016/j.jcp.2023.112598

(two of our documents cited)

Carrillo, J.A., Dębiec, T., Gwiazda, P., Świerczewska-Gwiazda, A.:

DISSIPATIVE MEASURE-VALUED SOLUTIONS TO THE EULER-POISSON EQUATION,

SIAM Journal on Mathematical Analysis, 2024, 56(1), pp. 304-335

DOI: 10.1137/22M1525983

Westdickenberg, M.:

Minimal Acceleration for the Multi-dimensional Isentropic Euler Equations,

Archive for Rational Mechanics and Analysis, 2023, 247(3), 35

DOI: 10.1007/s00205-023-01864-x

Lanthaler, S., Mishra, S., Parés-Pulido, C.:

Statistical solutions of the incompressible Euler equations,

Mathematical Models and Methods in Applied Sciences, 2021, 31(2), pp. 223–292

DOI: 10.1142/S0218202521500068

Breit, D., Mensah, P.R.:

An Incompressible Polymer Fluid Interacting with a Koiter Shell,

Journal of Nonlinear Science, 2021, 31(1), 25

DOI: 10.1007/s00332-021-09678-5

01/2024 - 12/2027 principal investigator

VEGA grant 1/0709/24 Numerical analysis in compressible fluid dynamics

07/2024 - 06/2028 team member

APVV-23-0039 Qualitative properties of evolution problems from science and technology

01/2024 - 12/2026 team member

GAČR Grant GA24-11034S Dissipative systems in fluid dynamics

01/2023 - 12/2024 team member

VEGA grant 1/0084/23 Qualitative properties of nonlinear differential equations of both integer and non-integer orders

01/2021 - 12/2023 team member

GAČR Grant GA21-02411S Solving ill posed problems in the dynamics of compressible fluids