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Guaranteed and fully computable two-sided bounds of Friedrichs' constant
Vejchodský, Tomáš
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.
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Smooth approximation and its application to some 1D problems
Segeth, Karel
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.
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Computational and analytical a posteriori error estimates for finite element methods
Segeth, Karel
The analytical a posteriori error estimates are oriented to the use in h-methods, are usually constructed only for lowest-order polynomial approximation, and often depend on unknown constatns or functions. In this review paper, we present several error estimation procedures for some particular linear partial differential problems with special regards to the needs of the hp-method. We compare the advantages and drawbacks of a posteriori error estimators including computational ones.
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Aposteriorní odhady chyby pro adaptivní metody konečných prvků
Segeth, Karel
While the classical a posteriori error estimates are oriented to the use in h-finite element methods the contemporary higher-order hp-methods usually require new approaches in a posteriori error estimation. We present examples of error estimation procedures for some model linear problems with a special regard to the needs of the hp-method. In the conclusion, we assess the advantages and drawbacks of a posteriori error estimates including computational error estimates (reference solutions).
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hp-metody konečných prvků adaptivní v prostoru i v čase: Přehled metodologie
Šolín, P. ; Segeth, Karel ; Doležel, I.
We present a new class of self-adaptive higher-order finite element methods (hp-FEM) which are free of analytical error estimates and thus work equally well for virtually all PDE problems ranging from simple linear elliptic equations to complex time-dependent nonlinear multiphysics coupled problems. The methodology was used to solve various types of problems. In this paper we use a nonlinear combustion problem for illustration.
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