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A note on tension spline
Segeth, Karel
2015 - anglický
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.
Klíčová slova:
smooth interpolation; tension spline; Fourier transform
Plné texty jsou dostupné v digitálním repozitáři Akademie Věd.
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 ...
Variability of Turing patterns in reaction-diffusion systems
Rybář, Vojtěch; Vejchodský, Tomáš
2014 - anglický
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.
Klíčová slova:
diffusion driven instability; uniqueness; number of solutions
Dokument je dostupný na externích webových stránkách.
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 ...
On simplicial red refinement in three and higher dimensions
Korotov, S.; Křížek, Michal
2013 - anglický
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.
Klíčová slova:
red refinement; finite element analysis
Dokument je dostupný na externích webových stránkách.
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 ...
Analytical solution of rotationally symmetric Stokes flow near corners
Burda, P.; Novotný, Jaroslav; Šístek, Jakub
2013 - anglický
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.
Klíčová slova:
Stokes problem; rotationally symmetric domains
Dokument je dostupný na externích webových stránkách.
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 ...
A direct solver for finite element matrices requiring O(N log N) memory places
Vejchodský, Tomáš
2013 - anglický
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.
Klíčová slova:
stiffness matrix; efficient
Dokument je dostupný na externích webových stránkách.
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 ...
On selection of interface weights in domain decomposition methods
Čertíková, M.; Šístek, Jakub; Burda, P.
2013 - anglický
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.
Klíčová slova:
BDDC method
Dokument je dostupný na externích webových stránkách.
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 ...
Smooth approximation of data with applications to interpolating and smoothings
Segeth, Karel
2013 - anglický
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.
Klíčová slova:
smooth approximation
Dokument je dostupný na externích webových stránkách.
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 ...
Guaranteed and fully computable two-sided bounds of Friedrichs' constant
Vejchodský, Tomáš
2013 - anglický
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.
Klíčová slova:
numerical methods; computing
Dokument je dostupný na externích webových stránkách.
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 ...
Smooth approximation and its application to some 1D problems
Segeth, Karel
2012 - anglický
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.
Klíčová slova:
smooth approximation
Dokument je dostupný na externích webových stránkách.
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 ...
Computing upper bounds on Friedrichs' constant
Vejchodský, Tomáš
2012 - anglický
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.
Klíčová slova:
Friedrichs' constant
Dokument je dostupný na externích webových stránkách.
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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