|
|
Programs and Algorithms of Numerical Mathematics 15
Vejchodský, Tomáš ; Chleboun, J. ; Přikryl, Petr ; Segeth, Karel ; Šístek, Jakub
The book contains papers presented at the international seminar Programs and algorithms of numerical Mathematics 15 (PANM 15), held in Dolni Maxov, Czech Republic, June 6-11, 2010. It is the fifteen volume in the series of the PANM proceedings. The topics of contributions include numerical methods for fluid flow modelling, the finite element method, a posteriori error estimates, topics from numerical linear atgebra, etc.
|
|
|
Complementarity - the way towards guaranteed error estimates
Vejchodský, Tomáš
This paper presents a review of the complementary technique with the emphasis on computable and guaranteed upper bounds of the approximation error. For simplicity, the approach is described on a numerical solution of the Poisson problem. We derive the complementary error bounds, prove their fundamentals properties, present the method of hypercircle, mention possible generalizations and show a couple of numerical examples.
|
|
|
Programy a algoritmy numerické matematiky 14
Chleboun, Jan ; Přikryl, Petr ; Segeth, Karel ; Vejchodský, Tomáš
The book contains papers presented at the international seminar Programs and algorithms of numerical Mathematics 14 (PANM 14), held in Dolni Maxov, Czech Republic, June 1-6, 2008. It is the fourteenth volume in the series of the PANM proceedings. The topics of contributions include numerical methods for fluid flow modelling, the finite element method, a posteriori error estimates, topics from numerical linear atgebra, etc.
|
|
|
Výpočetní srovnání diskretizační a iterační chyby
Vejchodský, Tomáš
The paper presents the Poisson equation and its solution by the finite element method. A numerical example is show, where besides the exact solution also the exact discrete solution is explicitely known. This allows to compare the discretization error and the error in the approximate solution of the system of linear algebraic system.
|
|
| |
|
|
Programy a algoritmy numerické matematiky 13
Chleboun, Jan ; Segeth, Karel ; Vejchodský, Tomáš
The book contains papers presented at the international conference Programs and Algorithms of Numerical Mathematics 13(PANM 13) held in Prague, Czech Republic, May 28-31, 2006, in honor of Ivo Babuska´s 80th birthday. It is the thirteenth volume in the series of the PANM proceedings. The topics of contributions include numerical methods for fluid flow modelling, the finite element method, a posteriori error estimates, topics from numerical linear algebra, etc.
|
|
|
Problém adaptivity v hp verzi metody konečných prvků
Vejchodský, Tomáš
The hp-version of the finite element method (hp-FEM) is extremely efficient numerical method for solving partial differential equations. In comparison with low-order methods it is capable to achieve exponential rate of convergence even though the solution has singularities and/or boundary/internal layers. To achieve the exponential convergence it is necessary to adapt both the geometry of the mesh and the polynomial degrees. However, the optimal hp-adaptive strategy is still unknown. This paper gives brief introduction into the problematic of hp-FEM and hp-adaptivity and emphasizes those parts that are not optimally solved yet.
|
|
|
Zlepšování podmíněnosti v hp verzi metody konečných prvků
Vejchodský, Tomáš ; Šolín, Pavel
We present the problem of choice of higher-order basis functions for hp-version of the finite element method (hp-FEM) with respect to the condition number of the resulting stiffness matrix. We compare numerically the condition numbers for several popular sets of basis functions. We show that the best conditioning have the basis functions that are orthogonal in the energetic sense on the reference element. Moreover, any two sets of basis functions with this property have identical condition number of both stiffnes and mass matrices.
|
|
|
Diskrétní Greenova funkce a princip maxima
Vejchodský, Tomáš ; Šolín, Pavel
In this paper the discrete Green´s function (DGF) is introduced and its fundamental properties are proven. Further it is indicated how to use these results to prove the discrete maximum principle for 1D Poisson equation discretized by the hp-FEM with pure Dirichlet or with mixed Dirichlet-Neumann boundary conditions and with piecewise constant coefficient
|
|
|
Rychlý a zaručený aposteriorní odhad chyby
Vejchodský, Tomáš
The equilibrated residual method and the method of hypercircle are popular methods for a posteriori error estimation for linear elliptic problems. Both these methods are intended to produce guaranteed upper bounds of the energy norm of the error, but the equilibrated residual method is guaranteed only theoretically. The disadvantage of the hypercircle method is its globality, hence slowness. The combination of these two methods leads to local, hence fast, and guaranteed a posteriori error estimator.
|
guest ::
