Meshless Collocation: Error Estimates with Application to Dynamical Systems

Giesl, Peter; Wendland, Holger

doi:10.1137/060658813

Cited by 71 publications

(86 citation statements)

References 13 publications

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“…has a negative orbital derivative. Such error estimates were derived, for example in (Giesl, 2007;Giesl and Wendland, 2007), see also (Narcowich et al, 2005;Wendland, 2005).…”

Section: Mesh-free Collocationmentioning

confidence: 99%

Analysing Dynamical Systems - Towards Computing Complete Lyapunov Functions

Argáez

Hafstein

Giesl

2017

Proceedings of the 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

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Abstract:Ordinary differential equations arise in a variety of applications, including e.g. climate systems, and can exhibit complicated dynamical behaviour. Complete Lyapunov functions can capture this behaviour by dividing the phase space into the chain-recurrent set, determining the long-time behaviour, and the transient part, where solutions pass through. In this paper, we present an algorithm to construct complete Lyapunov functions. It is based on mesh-free numerical approximation and uses the failure of convergence in certain areas to determine the chain-recurrent set. The algorithm is applied to three examples and is able to determine attractors and repellers, including periodic orbits and homoclinic orbits.

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“…has a negative orbital derivative. Such error estimates were derived, for example in (Giesl, 2007;Giesl and Wendland, 2007), see also (Narcowich et al, 2005;Wendland, 2005).…”

Section: Mesh-free Collocationmentioning

confidence: 99%

Analysing Dynamical Systems - Towards Computing Complete Lyapunov Functions

Argáez

Hafstein

Giesl

2017

Proceedings of the 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

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“…The idea of a grid-free method of radial basis functions (refer, for example, to [14], [15] and the bibliography on RBF presented there) is as follows: we cover the closure of the domain Ω = Ω Γ ∪ by principal points j j (x , y ), j 1,n. = In this case, any two points cannot coincide.…”

Section: Idea Of a Numerical Methodsmentioning

confidence: 99%

“…Among such methods, we shall note the finite element method (FEM) [8,11] and a large group of grid-free methods, which include, for example, the method of radial basis functions (RBF) [11][12][13][14][15]. Note that the scientific literature reports the research into accuracy of the RBF method only for certain boundary problems (for example, articles [14,15]). At present, the only research tool of boundary problems is a computational experiment -comparing the solutions by the RBF method with the solutions by other methods, sufficiently substantiated theoretically (for example, FEM), as was demonstrated in paper [11].…”

Section: Modeling and Computer Simulation Of The Filtration Processesmentioning

confidence: 99%

Mathematical modeling and computer simulation of the filtration processes in earth dams

Ivanchuk

Martynyuk

Tsvetkova

et al. 2017

EEJET

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Побудовано математичну модель фільтра-ційної консолідації тіла греблі з водоводом і зоною вимиву в двовимірному випадку. Враховано вплив техногенних факторів. Розроблено комп'ю-терну програму для автоматизації розрахун-ків чисельного розв'язання крайової задачі без-сітковим методом радіальних базисних функцій. Проведено ряд числових експериментів і здійсне-ний їх аналіз Ключові слова: гідроенергетика, проблема фільтраційної консолідації, осмотичні явища, вільна поверхня, об'єктно-орієнтоване програ-мування Построена математическая модель фильтра-ционной консолидации тела плотины с водово-дом и зоной размыва в двумерном случае. Учтено влияние техногенных факторов. Разработана компьютерная программа для автоматиза-ции расчетов численного решения краевой зада-чи беcсеточным методом радиальных базисных функций. Проведен ряд числовых экспериментов и осуществлен их анализ Ключевые слова: гидроэнергетика, проблема фильтрационной консолидации, осмотические явления, свободная поверхность, объектно-ори-ентированное программирование UDC 519.61/.64:627.05

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“…This has been done in [8] for the special case of ∂Ω being the sphere in R d and in a more general context in [9].…”

Section: All Of This Together With Proposition 31 Givesmentioning

confidence: 99%

A high-order, analytically divergence-free discretization method for Darcy’s problem

Schräder

Wendland

2010

Math. Comp.

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Abstract. We develop and analyze a meshfree discretization method for Darcy's problem. Our approximation scheme is based upon optimal recovery which leads to a collocation scheme using divergence-free positive definite kernels. Besides producing analytically incompressible flow fields, our method can be of arbitrary order, works in arbitrary space dimension and for arbitrary geometries. After deriving the scheme, we investigate the approximation error for smooth target functions and derive optimal approximation orders. Finally, we illustrate the method by giving numerical examples.

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Meshless Collocation: Error Estimates with Application to Dynamical Systems

Cited by 71 publications

References 13 publications

Analysing Dynamical Systems - Towards Computing Complete Lyapunov Functions

Analysing Dynamical Systems - Towards Computing Complete Lyapunov Functions

Mathematical modeling and computer simulation of the filtration processes in earth dams

A high-order, analytically divergence-free discretization method for Darcy’s problem

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