Currently displaying 1 – 20 of 22

Showing per page

Order by Relevance | Title | Year of publication

On the quality of local flux reconstructions for guaranteed error bounds

Vejchodský, Tomáš — 2015

Application of Mathematics 2015

In this contribution we consider elliptic problems of a reaction-diffusion type discretized by the finite element method and study the quality of guaranteed upper bounds of the error. In particular, we concentrate on complementary error bounds whose values are determined by suitable flux reconstructions. We present numerical experiments comparing the performance of the local flux reconstruction of Ainsworth and Vejchodsky [2] and the reconstruction of Braess and Schöberl [5]. We evaluate the efficiency...

A direct solver for finite element matrices requiring O ( N log N ) memory places

Vejchodský, Tomáš — 2013

Applications of Mathematics 2013

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,...

Computing upper bounds on Friedrichs’ constant

Vejchodský, Tomáš — 2012

Applications of Mathematics 2012

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 p r i o r i - a p o s t e r i o r i i n e q u a l i t i e s [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...

Fast and guaranteed a posteriori error estimator

Vejchodský, Tomáš — 2004

Programs and Algorithms of Numerical Mathematics

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....

Guaranteed and fully computable two-sided bounds of Friedrichs’ constant

Vejchodský, Tomáš — 2013

Programs and Algorithms of Numerical Mathematics

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.

Complementarity - the way towards guaranteed error estimates

Vejchodský, Tomáš — 2010

Programs and Algorithms of Numerical Mathematics

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 fundamental properties, present the method of hypercircle, mention possible generalizations and show a couple of numerical examples.

On the number of stationary patterns in reaction-diffusion systems

Rybář, VojtěchVejchodský, Tomáš — 2015

Application of Mathematics 2015

We study systems of two nonlinear reaction-diffusion partial differential equations undergoing diffusion driven instability. Such systems may have spatially inhomogeneous stationary solutions called Turing patterns. These solutions are typically non-unique and it is not clear how many of them exists. Since there are no analytical results available, we look for the number of distinct stationary solutions numerically. As a typical example, we investigate the reaction-diffusion system designed to model...

Discrete Green's function and maximum principles

Vejchodský, Tomᚊolín, Pavel — 2006

Programs and Algorithms of Numerical Mathematics

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 h p -FEM with pure Dirichlet or with mixed Dirichlet-Neumann boundary conditions and with piecewise constant coefficient.

Irregularity of Turing patterns in the Thomas model with a unilateral term

Rybář, VojtěchVejchodský, Tomáš — 2015

Programs and Algorithms of Numerical Mathematics

In this contribution we add a unilateral term to the Thomas model and investigate the resulting Turing patterns. We show that the unilateral term yields nonsymmetric and irregular patterns. This contrasts with the approximately symmetric and regular patterns of the classical Thomas model. In addition, the unilateral term yields Turing patterns even for smaller ratio of diffusion constants. These conclusions accord with the recent findings about the influence of the unilateral term in a model for...

The discrete maximum principle for Galerkin solutions of elliptic problems

Tomáš Vejchodský — 2012

Open Mathematics

This paper provides an equivalent characterization of the discrete maximum principle for Galerkin solutions of general linear elliptic problems. The characterization is formulated in terms of the discrete Green’s function and the elliptic projection of the boundary data. This general concept is applied to the analysis of the discrete maximum principle for the higher-order finite elements in one-dimension and to the lowest-order finite elements on simplices of arbitrary dimension. The paper surveys...

Comparison principle for a nonlinear parabolic problem of a nonmonotone type

Tomas Vejchodský — 2002

Applicationes Mathematicae

A nonlinear parabolic problem with the Newton boundary conditions and its weak formulation are examined. The problem describes nonstationary heat conduction in inhomogeneous and anisotropic media. We prove a comparison principle which guarantees that for greater data we obtain, in general, greater weak solutions. A new strategy of proving the comparison principle is presented.

Accurate reduction of a model of circadian rhythms by delayed quasi-steady state assumptions

Tomáš Vejchodský — 2014

Mathematica Bohemica

Quasi-steady state assumptions are often used to simplify complex systems of ordinary differential equations in the modelling of biochemical processes. The simplified system is designed to have the same qualitative properties as the original system and to have a small number of variables. This enables to use the stability and bifurcation analysis to reveal a deeper structure in the dynamics of the original system. This contribution shows that introducing delays to quasi-steady state assumptions...

Page 1 Next

Download Results (CSV)