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On nonoverlapping domain decomposition methods for the incompressible Navier-Stokes equations

Xuejun Xu, C. O. Chow, S. H. Lui (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, a Dirichlet-Neumann substructuring domain decomposition method is presented for a finite element approximation to the nonlinear Navier-Stokes equations. It is shown that the Dirichlet-Neumann domain decomposition sequence converges geometrically to the true solution provided the Reynolds number is sufficiently small. In this method, subdomain problems are linear. Other version where the subdomain problems are linear Stokes problems is also presented.

On numerical solution of compressible flow in time-dependent domains

Miloslav Feistauer, Jaromír Horáček, Václav Kučera, Jaroslava Prokopová (2012)

Mathematica Bohemica

The paper deals with numerical simulation of a compressible flow in time-dependent 2D domains with a special interest in medical applications to airflow in the human vocal tract. The mathematical model of this process is described by the compressible Navier-Stokes equations. For the treatment of the time-dependent domain, the arbitrary Lagrangian-Eulerian (ALE) method is used. The discontinuous Galerkin finite element method (DGFEM) is used for the space semidiscretization of the governing equations...

On power series solutions for the Euler equation, and the Behr–Nečas–Wu initial datum

Carlo Morosi, Mario Pernici, Livio Pizzocchero (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We consider the Euler equation for an incompressible fluid on a three dimensional torus, and the construction of its solution as a power series in time. We point out some general facts on this subject, from convergence issues for the power series to the role of symmetries of the initial datum. We then turn the attention to a paper by Behr, Nečas and Wu, ESAIM: M2AN 35 (2001) 229–238; here, the authors chose a very simple Fourier polynomial as an initial datum for the Euler equation and analyzed...

On some Boussinesq systems in two space dimensions: theory and numerical analysis

Vassilios A. Dougalis, Dimitrios E. Mitsotakis, Jean-Claude Saut (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

A three-parameter family of Boussinesq type systems in two space dimensions is considered. These systems approximate the three-dimensional Euler equations, and consist of three nonlinear dispersive wave equations that describe two-way propagation of long surface waves of small amplitude in ideal fluids over a horizontal bottom. For a subset of these systems it is proved that their Cauchy problem is locally well-posed in suitable Sobolev classes. Further, a class of these systems is discretized...

On some implicit and semi-implicit staggered schemes for the shallow water and Euler equations

R. Herbin, W. Kheriji, J.-C. Latché (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper, we propose implicit and semi-implicit in time finite volume schemes for the barotropic Euler equations (hence, as a particular case, for the shallow water equations) and for the full Euler equations, based on staggered discretizations. For structured meshes, we use the MAC finite volume scheme, and, for general mixed quadrangular/hexahedral and simplicial meshes, we use the discrete unknowns of the Rannacher−Turek or Crouzeix−Raviart finite elements. We first show that a solution...

On stability of the P n mod / P n element for incompressible flow problems

Petr Knobloch (2006)

Applications of Mathematics

It is well known that finite element spaces used for approximating the velocity and the pressure in an incompressible flow problem have to be stable in the sense of the inf-sup condition of Babuška and Brezzi if a stabilization of the incompressibility constraint is not applied. In this paper we consider a recently introduced class of triangular nonconforming finite elements of n th order accuracy in the energy norm called P n elements. For n ≤ 3 we show that the stability condition holds if the velocity...

On the Charney Conjecture of Data Assimilation Employing Temperature Measurements Alone: The Paradigm of 3D Planetary Geostrophic Model

Aseel Farhat, Evelyn Lunasin, Edriss S. Titi (2016)

Mathematics of Climate and Weather Forecasting

Analyzing the validity and success of a data assimilation algorithmwhen some state variable observations are not available is an important problem in meteorology and engineering. We present an improved data assimilation algorithm for recovering the exact full reference solution (i.e. the velocity and temperature) of the 3D Planetary Geostrophic model, at an exponential rate in time, by employing coarse spatial mesh observations of the temperature alone. This provides, in the case of this paradigm,...

On the connection between some Riemann-solver free approaches to the approximation of multi-dimensional systems of hyperbolic conservation laws

Tim Kröger, Sebastian Noelle, Susanne Zimmermann (2004)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper, we present some interesting connections between a number of Riemann-solver free approaches to the numerical solution of multi-dimensional systems of conservation laws. As a main part, we present a new and elementary derivation of Fey’s Method of Transport (MoT) (respectively the second author’s ICE version of the scheme) and the state decompositions which form the basis of it. The only tools that we use are quadrature rules applied to the moment integral used in the gas kinetic derivation...

On the connection between some Riemann-solver free approaches to the approximation of multi-dimensional systems of hyperbolic conservation laws

Tim Kröger, Sebastian Noelle, Susanne Zimmermann (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we present some interesting connections between a number of Riemann-solver free approaches to the numerical solution of multi-dimensional systems of conservation laws. As a main part, we present a new and elementary derivation of Fey's Method of Transport (MoT) (respectively the second author's ICE version of the scheme) and the state decompositions which form the basis of it. The only tools that we use are quadrature rules applied to the moment integral used in the...

On the domain geometry dependence of the LBB condition

Evgenii V. Chizhonkov, Maxim A. Olshanskii (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The LBB condition is well-known to guarantee the stability of a finite element (FE) velocity - pressure pair in incompressible flow calculations. To ensure the condition to be satisfied a certain constant should be positive and mesh-independent. The paper studies the dependence of the LBB condition on the domain geometry. For model domains such as strips and rings the substantial dependence of this constant on geometry aspect ratios is observed. In domains with highly anisotropic substructures...

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