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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 ...
Spectral methods for reaction-diffusion systems
Rybář, Vojtěch
2013 - anglický
Although spectral methods proved to be numerical methods that can significantly speed up the computation of solutions of systems of reaction-diffusion equations, finite difference and finite element methods still prevail as the most widespread methods. This contribution offers comparison of the performance of the Fourier spectral method with finite element method for reaction-diffusion system modeling the generation of pigment patterns on the coat of the leopard.
Klíčová slova:
reaction-diffusion system
Plné texty jsou dostupné na jednotlivých ústavech Akademie věd ČR.
Spectral methods for reaction-diffusion systems
Although spectral methods proved to be numerical methods that can significantly speed up the computation of solutions of systems of reaction-diffusion equations, finite difference and finite element ...
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]. ...
Analytical solution of Stokes flow near corners and applications to numerical solution of Navier-Stokes equations with high precision
Burda, P.; Novotný, Jaroslav; Šístek, Jakub
2012 - anglický
We present analytical solution of the Stokes problem in 2D domains. This is then used to find the asymptotic behavior of the solution in the vicinity of corners, also for Navier-Stokes equations in 2D. We apply this to construct very precise numerical finite element solution.
Klíčová slova:
Stokes flow; singularities; corners; asymptotic solution
Dokument je dostupný na externích webových stránkách.
Analytical solution of Stokes flow near corners and applications to numerical solution of Navier-Stokes equations with high precision
We present analytical solution of the Stokes problem in 2D domains. This is then used to find the asymptotic behavior of the solution in the vicinity of corners, also for Navier-Stokes equations in ...
Numerical comparison of different choices of interface weights in the BDDC method
Čertíková, M.; Burda, P.; Šístek, Jakub
2012 - anglický
Balancing Domain Decomposition by Constraints (BDDC) belongs to the class of primal substructuring Domain Decomposition (DD) methods. DD methods are iterative methods successfully used in engineering to parallelize solution of large linear systems arising from discretization of second order elliptic problems. Substructuring DD methods represent an important class of DD methods. Their main idea is to divide the underlying domain into nonoverlapping subdomains and solvemany relatively small, local problems on subdomains instead of one large problem on the whole domain. In primal methods, it has to be specified how to distribute interface residuals among subdomains and how to obtain global, interface values of solution from local values on adjacent subdomains. Usually a weighted average is used with some simple choice of weights.
Klíčová slova:
BDDC; domain decomposition; iterative substructuring; averaging
Dokument je dostupný na externích webových stránkách.
Numerical comparison of different choices of interface weights in the BDDC method
Balancing Domain Decomposition by Constraints (BDDC) belongs to the class of primal substructuring Domain Decomposition (DD) methods. DD methods are iterative methods successfully used in engineering ...
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