**90**

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**On the problem of singular limits in a model of radiative flow**

Ducomet, B.; Nečasová, Šárka

2015 - English
We consider a "semi-relativistic" model of radiative viscous compressible Navier-Stokes-Fourier system coupled to the radiative transfer equation extending the classical model introduced in [8] and we study diffusion limits in the case of well-prepared initial data and Dirichlet boundary condition for the velocity field.
Keywords:
*radiation hydrodynamics; Navier-Stokes-Fourier system; weak solution*
Fulltext is available at external website.
On the problem of singular limits in a model of radiative flow

We consider a "semi-relativistic" model of radiative viscous compressible Navier-Stokes-Fourier system coupled to the radiative transfer equation extending the classical model introduced in [8] and we ...

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**Some notes about the homogenization problem for the Navier-Stokes equations**

Feireisl, Eduard; Namlyeyeva, Yuliya; Nečasová, Šárka

2015 - English
We study the homogenization problem for the evolutionary Navier-Stokes system under the critical size of obstacles. Convergence towards the limit system of Brinkman's type is shown under very mild assumptions concerning the shape of the obstacles and their mutual distance.
Keywords:
*Navier-Stokes system; homogenization; Brinkman's law*
Fulltext is available at external website.
Some notes about the homogenization problem for the Navier-Stokes equations

We study the homogenization problem for the evolutionary Navier-Stokes system under the critical size of obstacles. Convergence towards the limit system of Brinkman's type is shown under very mild ...

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**Different approaches to interface weights in the BDDC method in 3D**

Čertíková, M.; Šístek, Jakub; Burda, P.

2015 - English
In this paper, we discuss the choice of weights in averaging of local (subdomain) solutions on the interface for the BDDC method (Balancing Domain Decomposition by Constraints). We try to find relations among different choices of the interface weights and compare them numerically on model problems of the Poisson equation and linear elasticity in 3D. Problems with jumps in coefficients of material properties are considered and both regular and irregular interfaces between subdomains are tested.
Keywords:
*BDDC; interface weights; coefficient jumps*
Available in digital repository of the ASCR
Different approaches to interface weights in the BDDC method in 3D

In this paper, we discuss the choice of weights in averaging of local (subdomain) solutions on the interface for the BDDC method (Balancing Domain Decomposition by Constraints). We try to find ...

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**Smooth approximation spaces based on a periodic system**

Segeth, Karel

2015 - English
A way of data approximation called smooth was introduced by Talmi and Gilat in 1977. Such an approach employs a (possibly infinite) linear combination of smooth basis functions with coefficients obtained as the unique solution of a minimization problem. While the minimization guarantees the smoothness of the approximant and its derivatives, the constraints represent the interpolating or smoothing conditions at nodes. In the contribution, a special attention is paid to the periodic basis system $exp(-ii kx)$. A 1D numerical example is presented.
Keywords:
*smooth interpolation; data interpolation; cubic spline interpolation*
Available in digital repository of the ASCR
Smooth approximation spaces based on a periodic system

A way of data approximation called smooth was introduced by Talmi and Gilat in 1977. Such an approach employs a (possibly infinite) linear combination of smooth basis functions with coefficients ...

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**Why quintic polynomial equations are not solvable in radicals**

Křížek, Michal; Somer, L.

2015 - English
We illustrate the main idea of Galois theory, by which roots of a polynomial equation of at least fifth degree with rational coefficients cannot general be expressed bz radicals, i.e., by the operations +, -, ., :, and .... Therefore, higher order polynomial equations are usually solved by approximate methods. They can also be solved algebraically by means of ultraradicals.
Keywords:
*Galois theory; finite group; permutation*
Available in digital repository of the ASCR
Why quintic polynomial equations are not solvable in radicals

We illustrate the main idea of Galois theory, by which roots of a polynomial equation of at least fifth degree with rational coefficients cannot general be expressed bz radicals, i.e., by the ...

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**Irregularity of turing patterns in the Thomas model with a unilateral term**

Rybář, Vojtěch; Vejchodský, Tomáš

2015 - English
In this contribution we add a unilateral term to the Thomas model and investigate the resulting Turing patterns. We show that the unilateral term yields nonsymmetric and irregular patterns. This contrasts with the approximately symmetric and regular patterns of the classical Thomas model. In addition, the unilateral term yields Turing patterns even for smaller ratio of diffusion constants. These conclusions accord with the recent findings about the influence of the unilateral term in a model for mammalian coat patterns. This indicates that the observed effects of the unilateral term are general and apply to a variety of systems.
Keywords:
*reactin-diffusion system; Thomas model; Turing instability*
Available in digital repository of the ASCR
Irregularity of turing patterns in the Thomas model with a unilateral term

In this contribution we add a unilateral term to the Thomas model and investigate the resulting Turing patterns. We show that the unilateral term yields nonsymmetric and irregular patterns. This ...

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**On the number of stationary patterns in reaction-diffusion systems**

Rybář, Vojtěch; Vejchodský, Tomáš

2015 - English
We study systems of two nonlinear reaction-diffusion partial differential equations undergoing diffusion driven instability. Such systems may have spatially inhomogeneous stationary solutions called Turing patterns. These solutions are typically non-unique and it is not clear how many of them exists. Since there are no analytical results available, we look for the number of distinct stationary solutions numerically. As a typical example, we investigate the reaction-diffusion systém designed to model coat patterns in leopard and jaguar.
Keywords:
*diffusion driven instability; Turing patterns; classification of non-unique solutions*
Available in digital repository of the ASCR
On the number of stationary patterns in reaction-diffusion systems

We study systems of two nonlinear reaction-diffusion partial differential equations undergoing diffusion driven instability. Such systems may have spatially inhomogeneous stationary solutions called ...

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**On the quality of local flux reconstructions for guaranteed error bounds**

Vejchodský, Tomáš

2015 - English
In this contribution we consider elliptic problems of a reaction-diffucion type discretized by the finite element method and study the quality of guaranteed upper bounds of the error. In particular, we concentrate on complementary error bounds whose values are determined by suitable flux reconstructions. We present numerical experiments comparing the performance of the local flux reconstruction of Ainsworth and Vejchodský [2] and the reconstruction of Braess and Schröberl [5]. We evaluate the efficiency of these flux reconstructions by their comparison with the optimal flux reconstruction computed as a global minimization problem.
Keywords:
*a posteriori error estimates; complementarity; index of effectivity; elliptic problem*
Fulltext is available at external website.
On the quality of local flux reconstructions for guaranteed error bounds

In this contribution we consider elliptic problems of a reaction-diffucion type discretized by the finite element method and study the quality of guaranteed upper bounds of the error. In particular, ...

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**A parallel finite element solver for unsteady incompressible Navier-Stokes equations**

Šístek, Jakub

2015 - English
A parallel solver for unsteady incompressible Navier-Stokes equations is presented. It is based on the finite element method combined with the pressure-correction approach. Semi-implicit treatment of the convective term is considered, leading to five systems of linear algebraic equations to be solved in each time step. Krylov subspace iterative methods are employed for the solution of these systems with a particular emphasis on efficient parallel preconditioners. A simulation of a benchmark problem of incompressible viscous flow around a sphere at Reynolds number 300 is presented and compared with literature.
Keywords:
*pressure-correction; parallel algorithms; flow around a sphere*
Fulltext is available at external website.
A parallel finite element solver for unsteady incompressible Navier-Stokes equations

A parallel solver for unsteady incompressible Navier-Stokes equations is presented. It is based on the finite element method combined with the pressure-correction approach. Semi-implicit treatment of ...

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**An application of the BDDC method to the Navier-Stokes equations in 3-D cavity**

Hanek, M.; Šístek, Jakub; Burda, P.

2015 - English
We deal with numerical simulation of incompressible flow governed by the Navier-Stokes equations. The problem is discretised using the finite element method, and the arising system of nonlinear equations is solved by Picard iteration. We explore the applicability of the Balancing Domain Decomposition by Constraints (BDDC) method to nonsymmetric problems arising from such linearisation. One step of BDDC is applied as the preconditioner for the stabilized variant of the biconjugate gradient (BiCGstab) method. We present results for a 3-D cavity problem computed on 32 cores of a parallel supercomputer.
Keywords:
*incompressible fluid; Navier-Stoken equations; BDDC; Taylor-Hood finite element*
Available in digital repository of the ASCR
An application of the BDDC method to the Navier-Stokes equations in 3-D cavity

We deal with numerical simulation of incompressible flow governed by the Navier-Stokes equations. The problem is discretised using the finite element method, and the arising system of nonlinear ...

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