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Programs and Algorithms of Numerical Mathematics 21 : Jablonec nad Nisou, June 19-24, 2022 : Proceedings of Seminar
Chleboun, J. ; Kůs, Pavel ; Papež, Jan ; Rozložník, Miroslav ; Segeth, Karel ; Šístek, Jakub
These proceedings contain peer-reviewed papers that are based on the invited lectures, short communications, and posters presented at the 21st seminar Programs and Algorithms of Numerical Mathematics (PANM) held in Merkur Hotel, Jablonec nad Nisou, Czech Republic, June 19-24, 2022.\nThe seminar was organized by the Institute of Mathematics of the Czech Academy of Sciences under the auspices of EU-MATHS-IN.CZ, Czech Network for Mathematics in Industry, and with the nancial support provided by the RSJ Foundation. It continued the previous seminars on mathematical software and numerical methods held (biennially, with only one exception) in Alšovice, Bratříkov, Janov nad Nisou, Kořenov, Lázně Libverda, Dolní Maxov, Prague, and Hejnice in the period 1983-2020. The objective of this series of seminars is to provide a forum for presenting and discussing advanced topics in numerical analysis, computer implementation of numerical algorithms, new approaches to mathematical modeling, and single- or multi-processor applications of computational methods.
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Quantum vacua, curved spacetime and singularities
Kůs, Pavel ; Iorio, Alfredo (advisor)
In this work we investigate the Weyl anomaly from a new perspective. Our goal is to identify a set-up for which the classical Weyl symmetry is not broken, at the quantum level by the usual arguments related to the Euler invariants, but rather by the impact of other geometrical obstructions. Therefore, we work, mostly, in three spatiotemporal dimensions, where general arguments guarantee the absence of trace anomalies. In par- ticular, our interest here is on whether various types of singularities, emerging in the description of the differential geometry of surfaces, could induce some form of quantum inequivalence, even though the classical symmetry is at work. To this end, we work with a very special three-dimensional metric, whose nontriviality is fully in its spatial two-dimensional part. The last ingredient we use, to clean-up the way from other com- plications, is to work with physical systems where no Weyl gauge field is necessary, to have the classical invariance. The system we focus on is then the massless Dirac field the- ory (that, as well known, enjoys local Weyl symmetry) in three-dimensional conformally flat spacetimes. With these premises, the research programme consists of three steps. The first step is to find the coordinate transformations that link the conformal factor identifying the...
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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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Numerical solution of convection-diffusion equations with the aid of adaptive time-space higher order methods
Kůs, Pavel ; Dolejší, Vít (advisor) ; Felcman, Jiří (referee)
This thesis deals with solution of scalar nonlinear convection-diffusion equation with aid of discontinuous Galerkin method. It's aim is to implement an adaptive choice of time step. To do this, we derived 2 sufficiently stable methods for solution of systems of ordinary differential equations obtained by space semidicretization, which is carried out by the discontinuous Galerkin method. Using those two approximate solutions, we estimate local error of discretization. Using it, we are able to choose following time step in such way, that local error is approximately equal to given tolerance. Several numerical simulations were carried out to check properties of this method.
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Quantum vacua, curved spacetime and singularities
Kůs, Pavel ; Iorio, Alfredo (advisor) ; Jakubský, Vít (referee)
In this work we investigate the Weyl anomaly from a new perspective. Our goal is to identify a set-up for which the classical Weyl symmetry is not broken, at the quantum level by the usual arguments related to the Euler invariants, but rather by the impact of other geometrical obstructions. Therefore, we work, mostly, in three spatiotemporal dimensions, where general arguments guarantee the absence of trace anomalies. In par- ticular, our interest here is on whether various types of singularities, emerging in the description of the differential geometry of surfaces, could induce some form of quantum inequivalence, even though the classical symmetry is at work. To this end, we work with a very special three-dimensional metric, whose nontriviality is fully in its spatial two-dimensional part. The last ingredient we use, to clean-up the way from other com- plications, is to work with physical systems where no Weyl gauge field is necessary, to have the classical invariance. The system we focus on is then the massless Dirac field the- ory (that, as well known, enjoys local Weyl symmetry) in three-dimensional conformally flat spacetimes. With these premises, the research programme consists of three steps. The first step is to find the coordinate transformations that link the conformal factor identifying the...
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Automatic hp-adaptivity on Meshes with Arbitrary-Level Hanging Nodes in 3D
Kůs, Pavel
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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Programs and Algorithms of Numerical Mathematics 20 : Hejnice, June 21-26, 2019 : proceedings of seminar
Chleboun, J. ; Kůs, Pavel ; Přikryl, Petr ; Rozložník, Miroslav ; Segeth, Karel ; Šístek, Jakub
This book comprises papers that originated from the invited lectures, survey lectures, short communications, and posters presented at the 20th seminar Programs and Algorithms of Numerical Mathematics (PANM) held in Hejnice, Czech Republic, June 21-26, 2020. All the papers have been peer-reviewed. The seminar was organized by the Institute of Mathematics of the Czech Academy of Sciences under the auspices of EU-MATHS-IN.cz, Czech Network for Mathematics in Industry. The objective of this series of seminars is to provide a forum for presenting and discussing advanced theoretical as well as practical topics in numerical analysis, computer implementation of algorithms, new approaches to mathematical modeling, and single- or multi-processor applications of computational methods.
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Conformal symmetry and vortices in graphene
Kůs, Pavel ; Iorio, Alfredo (advisor) ; Jizba, Petr (referee)
This study provides an introductory insight into the complex field of graphene and its relativistic-like behaviour. The thesis is opened by an overview to this topic and draws special attention to interesting non-topological vortex solutions of the Liouville equation found by P. A. Horváthy and J.-C. Yéra, which emerge in a context of the Chern-Simons theory [1], [2] and have been put into context of graphene [3], [4]. We introduce the massless Dirac field theory, well describing electronic properties of graphene in the low energy limit, and point to the fact that the action of the massless Dirac field is invariant under Weyl transformations, which has far-reaching consequences. When the graphene membrane is suitably deformed, we assume that the correct description is that of a Dirac field on a curved spacetime. In particular, an important case is that of conformally flat 2+1-dimensional spacetimes. These are obtained when the spatial part of the metric describes a surface of constant intrinsic curvature [3]. In other words, the conformal factor of such spatial metrics has to satisfy the Liouville equation, an important equation of mathematical physics. In this work, we have identified the kind of surfaces to which the Horváthy- Yéra conformal factors, above recalled, correspond, and have provided...
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Programs and Algorithms of Numerical Mathematics 18 : Janov nad Nisou, June 19-24, 2016 : proceedings of seminar
Chleboun, J. ; Kůs, Pavel ; Přikryl, Petr ; Segeth, Karel ; Šístek, Jakub ; Vejchodský, Tomáš
This book comprises papers that originated from the invited lectures, survey lectures, short communications, and posters presented at the 18th seminar Programs and Algorithms of Numerical Mathematics (PANM) held in Janov nad Nisou, Czech Republic, June 19-24, 2016. All the papers have been peer-reviewed. The seminar was organized by the Institute of Mathematics of the Czech Academy of Sciences under the auspices of EU-MATHS-IN.cz, Czech Network for Mathematics in Industry. It continued the previous seminars on mathematical software and numerical methods held (biennially, with only one exception) in Al šovice, Bratří kov, Janov nad Nisou, Ko řenov, L ázně Libverda, Dolní Maxov, and Prague in the period 1983-2014. The objective of this series of seminars is to provide a forum for presenting and discussing advanced theoretical as well as practical topics in numerical analysis, computer implementation of algorithms, new approaches to mathematical modeling, and single- or multi-processor applications of computational methods.
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Programs and Algorithms of Numerical Mathematics 19 : Hejnice, June 24-29, 2018 : proceedings of seminar
Chleboun, J. ; Kůs, Pavel ; Přikryl, Petr ; Rozložník, Miroslav ; Segeth, Karel ; Šístek, Jakub ; Vejchodský, Tomáš
These proceedings contain peer-reviewed papers that are based on the invited lectures, survey lectures, short communications, and posters presented at the 19th seminar Programs and Algorithms of Numerical Mathematics (PANM) held in the International Center for Spiritual Rehabilitation, Hejnice, Czech Republic, June 24-29, 2018. The seminar was organized by the Institute of Mathematics of the Czech Academy of Sciences under the auspices of EU-MATHS-IN.cz, Czech Network for Mathematics in Industry, and with the financial support provided by the RSJ Foundation. It continued the previous seminars on mathematical software and numerical methods held (biennially, with only one exception) in Alšovice, Bratříkov, Janov nad Nisou, Kořenov, Lázně Libverda, Dolní Maxov, and Prague in the period 1983-2016. The objective of this series of seminars is to provide a forum for presenting and discussing advanced topics in numerical analysis, computer implementation of numerical algorithms, new approaches to mathematical modeling, and single- or multi-processor applications of computational methods.
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