**90**

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**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 ...

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**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 ...

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**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 ...

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**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 ...

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**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 ...

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**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 ...

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**Smooth approximation of data with applications to interpolating and smoothings**

Segeth, Karel

2013 - English
In the paper, we are concerned with some computational aspects of smooth approximation of data. This approach to approximation employs a (possibly infinite) linear combinations of smooth functions with coefficients obtained as the solution of a variational problem, where constraints represent the conditions of interpolating or smoothing. Some 1D numerical examples are presented.
Keywords:
*smooth approximation*
Fulltext is available at external website.
Smooth approximation of data with applications to interpolating and smoothings

In the paper, we are concerned with some computational aspects of smooth approximation of data. This approach to approximation employs a (possibly infinite) linear combinations of smooth functions ...

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**Guaranteed and fully computable two-sided bounds of Friedrichs' constant**

Vejchodský, Tomáš

2013 - English
This contribution presents a general numerical method for computing lower and upper bound of the optimal constant in Friedrichs’ inequality. The standard Rayleigh-Ritz method is used for the lower bound and the method of a priori-a posteriori inequalities is employed for the upper bound. Several numerical experiments show applicability and accuracy of this approach.
Keywords:
*numerical methods; computing*
Fulltext is available at external website.
Guaranteed and fully computable two-sided bounds of Friedrichs' constant

This contribution presents a general numerical method for computing lower and upper bound of the optimal constant in Friedrichs’ inequality. The standard Rayleigh-Ritz method is used for the lower ...

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**Smooth approximation and its application to some 1D problems**

Segeth, Karel

2012 - English
In the contribution, we are concerned with the exact interpolation of the data at nodes given and also with the smoothness of the interpolating curve and its derivatives. This task is called the problem of smooth approximation of data. The interpolating curve or surface is defined as the solution of a variational problem with constraints. We discuss the proper choice of basis systems for this way of approximation and present the results of several 1D numerical examples that show the quality of smooth approximation.
Keywords:
*smooth approximation*
Fulltext is available at external website.
Smooth approximation and its application to some 1D problems

In the contribution, we are concerned with the exact interpolation of the data at nodes given and also with the smoothness of the interpolating curve and its derivatives. This task is called the ...

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**Computing upper bounds on Friedrichs' constant**

Vejchodský, Tomáš

2012 - English
This contribution shows how to compute upper bounds of the optimal constant in Friedrichs’ and similar inequalities. The approach is based on the method of a prioria posteriori inequalities [9]. However, this method requires trial and test functions with continuous second derivatives. We show how to avoid this requirement and how to compute the bounds on Friedrichs’ constant using standard finite element methods. This approach is quite general and allows variable coefficients and mixed boundary conditions. We use the computed upper bound on Friedrichs’ constant in a posteriori error estimation to obtain guaranteed error bounds.
Keywords:
*Friedrichs' constant*
Fulltext is available at external website.
Computing upper bounds on Friedrichs' constant

This contribution shows how to compute upper bounds of the optimal constant in Friedrichs’ and similar inequalities. The approach is based on the method of a prioria posteriori inequalities [9]. ...

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