Number of found documents: 76
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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 ...

Rybář, Vojtěch; Vejchodský, Tomáš
Matematický ústav, 2015

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, ...

Vejchodský, Tomáš
Matematický ústav, 2015

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 ...

Hanek, M.; Šístek, Jakub; Burda, P.
Matematický ústav, 2015

A note on tension spline
Segeth, Karel
2015 - English
Spline theory is mainly grounded on two approaches: the algebraic one (where splines are understood as piecewise smooth functions) and the variational one (where splines are obtained via minimization of quadratic functionals with constraints). We show that the general variational approach called smooth interpolation introduced by Talmi and Gilat covers not only the cubic spline but also the well known tension spline (called also spline in tension or spline with tension). We present the results of a 1D numerical example that show the advantages and drawbacks of the tension spline. Keywords: smooth interpolation; tension spline; Fourier transform Available in digital repository of the ASCR
A note on tension spline

Spline theory is mainly grounded on two approaches: the algebraic one (where splines are understood as piecewise smooth functions) and the variational one (where splines are obtained via minimization ...

Segeth, Karel
Matematický ústav, 2015

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 ...

Šístek, Jakub
Matematický ústav, 2015

Variability of Turing patterns in reaction-diffusion systems
Rybář, Vojtěch; Vejchodský, Tomáš
2014 - English
The paper presents a result about the number of distinct stationary solutions of a reaction-diffusion system exhibing the Turing instability. Relative frequency of observed solutions as they evolve from random initial conditions is analysed as well. Keywords: diffusion driven instability; uniqueness; number of solutions Fulltext is available at external website.
Variability of Turing patterns in reaction-diffusion systems

The paper presents a result about the number of distinct stationary solutions of a reaction-diffusion system exhibing the Turing instability. Relative frequency of observed solutions as they evolve ...

Rybář, Vojtěch; Vejchodský, Tomáš
Matematický ústav, 2014

On simplicial red refinement in three and higher dimensions
Korotov, S.; Křížek, Michal
2013 - English
We show that in dimensions higher than two, the popular “red refinement” tech- nique, commonly used for simplicial mesh refinements and adaptivity in the finite element analysis and practice, never yields subsimplices which are all acute even for an acute father element as opposed to the two-dimensional case. In the three-dimensional case we prove that there exists only one tetrahedron that can be partitioned by red refinement into eight congruent subtetrahedra that are all similar to the original one. Keywords: red refinement; finite element analysis Fulltext is available at external website.
On simplicial red refinement in three and higher dimensions

We show that in dimensions higher than two, the popular “red refinement” tech- nique, commonly used for simplicial mesh refinements and adaptivity in the finite element analysis and practice, never ...

Korotov, S.; Křížek, Michal
Matematický ústav, 2013

Analytical solution of rotationally symmetric Stokes flow near corners
Burda, P.; Novotný, Jaroslav; Šístek, Jakub
2013 - English
We present analytical solution of the Stokes problem in rotationally symmetric domains. This is then used to find the asymptotic behaviour of the solution in the vicinity of corners, also for Navier-Stokes equations. We apply this to construct very precise numerical finite element solution. Keywords: Stokes problem; rotationally symmetric domains Fulltext is available at external website.
Analytical solution of rotationally symmetric Stokes flow near corners

We present analytical solution of the Stokes problem in rotationally symmetric domains. This is then used to find the asymptotic behaviour of the solution in the vicinity of corners, also for ...

Burda, P.; Novotný, Jaroslav; Šístek, Jakub
Matematický ústav, 2013

A direct solver for finite element matrices requiring O(N log N) memory places
Vejchodský, Tomáš
2013 - English
We present a method that in certain sense stores the inverse of the stiffness matrix in O(N log N) memory places, where N is the number of degrees of freedom and hence the matrix size. The setup of this storage format requires O(N^(3/2)) arithmetic operations. However, once the setup is done, the multiplication of the inverse matrix and a vector can be performed with O(N log N) operations. This approach applies to the first order finite element discretization of linear elliptic and parabolic problems in triangular domains, but it can be generalized to higher-order elements, variety of problems, and general domains. The method is based on a special hierarchical enumeration of vertices and on a hierarchical elimination of suitable degrees of freedom. Therefore, we call it hierarchical condensation of degrees of freedom. Keywords: stiffness matrix; efficient Fulltext is available at external website.
A direct solver for finite element matrices requiring O(N log N) memory places

We present a method that in certain sense stores the inverse of the stiffness matrix in O(N log N) memory places, where N is the number of degrees of freedom and hence the matrix size. The setup of ...

Vejchodský, Tomáš
Matematický ústav, 2013

On selection of interface weights in domain decomposition methods
Čertíková, M.; Šístek, Jakub; Burda, P.
2013 - English
Different choices of the averaging operator within the BDDC method are compared on a series of 2D experiments. Subdomains with irregular interface and with jumps in material coefficients are included into the study. Two new approaches are studied along three standard choices. No approach is shown to be universally superior to others, and the resulting recommendation is that an actual method should be chosen based on properties of the problem. Keywords: BDDC method Fulltext is available at external website.
On selection of interface weights in domain decomposition methods

Different choices of the averaging operator within the BDDC method are compared on a series of 2D experiments. Subdomains with irregular interface and with jumps in material coefficients are included ...

Čertíková, M.; Šístek, Jakub; Burda, P.
Matematický ústav, 2013

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