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Automatic hp-adaptivity on Meshes with Arbitrary-Level Hanging Nodes in 3D
Kůs, Pavel ; Vejchodský, Tomáš (advisor) ; Segeth, Karel (referee) ; Dolejší, Vít (referee)
The thesis is concerned with theoretical and practical aspects of the hp- adaptive finite element method for solving elliptic and electromagnetic prob- lems described by partial differential equations in three spatial dimensions. Besides the standard element refinements, the hp-adaptivity allows indepen- dent adaptation of degrees of the polynomial approximation as well. This leads to exponentially fast convergence even for problems with singularities. The efficiency of the hp-adaptivity is enhanced even more by the ability of the algorithm to work with meshes with arbitrary-level hanging nodes. This generality, however, leads to great complexity of the implementation. There- fore, the thesis concentrates on the mathematical analysis of algorithms that have led to successful implementation of the method. In addition, the the- sis discusses the numerical integration in 3D and the implementation of the method itself. Finally, numerical results obtained by this new implemen- tation are presented. They confirm advantages of hp-adaptivity on meshes with arbitrary-level hanging nodes. 1
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Diskrétní princip maxima v metodě konečných prvků prvního řádu
Klejchová, Martina ; Vejchodský, Tomáš (advisor) ; Knobloch, Petr (referee)
In this thesis we study the discrete maximum principle for a diffusion-reaction problem solved by means of various types of nite elements. The work includes the brief characterization of the problem and of the nite element method, de nition of the discrete maximum principle and general conditions for its validity. The main focus of this work is in the analysis of such geometric conditions for the shape and possibly also for the size of the speci c nite elements that guarantee the validity of the discrete maximum principle. Namely, we analyze conditions for intervals, triangles, rectangles, tetrahedra, blocks and triangular prisms.
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Konečné prvky v elektromagnetismu kompatibilní s De Rhamovým diagramem
Rybář, Vojtěch ; Doležel, Ivo (advisor) ; Vejchodský, Tomáš (referee)
Title: Finite elements for electromagnetics compatible with de Rham di- agram Author: Vojtěch Rybář Department: Department of Numerical Mathematics Supervisor: prof. Ing. Ivo Doležel, CSc. Abstract: The present work is devoted to the lowest-order finite elements for solving time-harmonic Maxwell's equations in two dimensions. Suc- cessful approximation of these equations requires the finite element spaces to be compatible with the de Rham diagram. However, the most often used basis functions (the Whitney functions) do not comply with this diagram. Therefore, we construct compatible bases and study their prop- erties. Since the construction is not unique, we investigate the influence of the particular choice on the conditioning of the corresponding finite element matrices. Finally, we utilize the special structure of the stiffness matrices, propose a few iterative schemes, and compare their convergence. Keywords: Maxwell's equations, edge finite element, de Rham diagram, finite element basis 1
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Computation of an anisotropic and nonlinear magnetic field by the finite element method
Kunický, Zdeněk ; Křížek, Michal (referee) ; Vejchodský, Tomáš (advisor)
In the present work we study the modelling of stationary magnetic fields in nonlinear anisotropic media by FEM. The magnetic characteristics of such materials are thoroughly examined and eventually applied to the construction of a full 2D model of an anisotropic steel sheet. Some improvements in the construction in comparison with the ones previously published are achieved. We also present an extension of a 3D model of steel and dielectric laminations for anisotropic sheets. We point out that the standard formulations and the subsequent theorems for the boundary value problems in fact do not correspond with the physical situation. Instead, we propose new formulations that reflect the real physical properties of matter. General existence and uniqueness theorems for the obtained boundary value problems are proved as well as the convergence theorems for the discrete solutions. Finally, the conventional and full 2D model of an anisotropic steel sheet are compared in two transformer core models using the adaptive Newton-Raphson iterative scheme and the obtained results are presented and analysed.
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Computational comparison of hp-adaptive approaches
Kubásek, Petr ; Feistauer, Miloslav (referee) ; Vejchodský, Tomáš (advisor)
Cílem této práce je porovnat řízení hp-adaptivního procesu pomocí referenčního řešení a různých aposteriorních odhadů chyby. Tyto přístupy jsou porovnávány z hlediska globální diskretizační chyby a potřebného počtu stupňů volnosti. Konkrétně se zabýváme explicitními residuálními odhady, implicitními residuálními odhady Dirichletova a Neumannova typu a hierarchickými odhady. Všechny odhady jsou v práci podrobně odvozeny včetně jejich nejvýznamnějších vlastností. Jednotlivé přístupy jsou srovnávány pomocí numerických experimentů. Na jejich základě lze ríci, že nejlepších výsledků dosahuje adaptivita řízená pomocí referenčního řešení společně s implicitním Dirichletovým odhadem. Referenční řešení se zdá být nejspolehlivější metodou zatímco implicitní Dirichletův odhad je, s výjimkou některých případů, nejrychlejší.
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Algebraic Error in Matrix Computations in the Context of Numerical Solution of Partial Differential Equations
Papež, Jan ; Strakoš, Zdeněk (advisor) ; Ramage, Alison (referee) ; Vejchodský, Tomáš (referee)
Title: Algebraic Error in Matrix Computations in the Context of Numerical Solution of Partial Differential Equations Author: Jan Papež Department: Department of Numerical Mathematics Supervisor: prof. Ing. Zdeněk Strakoš, DrSc., Department of Numerical Mathe- matics Abstract: Solution of algebraic problems is an inseparable and usually the most time-consuming part of numerical solution of PDEs. Algebraic computations are, in general, not exact, and in many cases it is even principally desirable not to perform them to a high accuracy. This has consequences that have to be taken into account in numerical analysis. This thesis investigates in this line some closely related issues. It focuses, in particular, on spatial distribution of the errors of different origin across the solution domain, backward error interpretation of the algebraic error in the context of function approximations, incorporation of algebraic errors to a posteriori error analysis, influence of algebraic errors to adaptivity, and construction of stopping criteria for (preconditioned) iterative algebraic solvers. Progress in these issues requires, in our opinion, understanding the interconnections between the phases of the overall solution process, such as discretization and algebraic computations. Keywords: Numerical solution of partial...
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Proceedings of the International Conference Applications of Mathematics 2015 : Prague, November 18-21, 2015
Brandts, J. ; Korotov, S. ; Křížek, Michal ; Segeth, Karel ; Šístek, Jakub ; Vejchodský, Tomáš
Professors Ivo Babuška, Milan Práger, and Emil Vitásek are renowned experts in numerical analysis and computational methods. Their fruitful scientific careers started in Prague, at the Institute of Mathematics of the Czechoslovak Academy of Sciences (now Czech Academy of Sciences). They collaborated there on various projects including the computational analysis of the construction technology for Orlík Dam. In 1966 they published their joint book entitled Numerical Processes in Differential Equations. It is an honor for the Institute of Mathematics to host a conference on the occasion of their birthdays.
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Programs and Algorithms of Numerical Mathematics 17 : Dolní Maxov, June 8-13, 2014 : Proceedings of Seminar
Chleboun, J. ; Přikryl, Petr ; Segeth, Karel ; Šístek, Jakub ; Vejchodský, Tomáš
This volume comprises peer-reviewed papers that are based on invited lectures, survey lectures, short communications, and posters presented at the 17th seminar Programs and Algorithms of Numerical Mathematics (PANM) held in Dolní Maxov, Czech Republic, June 8–13, 2014. The seminar was organized by the Institute of Mathematics of the Academy of Sciences of the Czech Republic. It continued the previous seminars on mathematical software and numerical methods held (with only one exception) biannually in\nAlšovice, Bratříkov, Janov nad Nisou, Kořenov, Lázně Libverda, Dolní Maxov, and Prague in the period 1983–2012. The objective of this series of seminars is to provide a forum for presenting and discussing advanced topics in numerical analysis, singleor multi-processor applications of computational methods, and new approaches to mathematical modeling.
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On the quality of local flux reconstructions for guaranteed error bounds
Vejchodský, Tomáš
In this contribution we consider elliptic problems of a reaction-diffucion 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 Vejchodský [2] and the reconstruction of Braess and Schröberl [5]. We evaluate the efficiency of these flux reconstructions by their comparison with the optimal flux reconstruction computed as a global minimization problem.
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