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Shadowing property of numerical methods for partial differential equations
Brichta, Ondřej ; Kučera, Václav (advisor) ; Knobloch, Petr (referee)
This thesis is focused on the shadowing property of numerical methods for partial differential equations. The goals of this thesis are the application of shadowing theory to the case of linear maps, modification of standard techniques for the purpose of this application and the application of the adapted theory to multistep schemes. In the in- troductory overview, we first focus on a study of the standard shadowing theory, then we formulate basic statements and prove a relationship between a conctractivity and shadowing. Afterwards we find the characterization of the shadowing property for linear maps. In the next sections, we adapt definitions of the shadowing theory to requirements of multistep methods. As an example, we apply the adapted theory to the Dufort-Frankel scheme in the third chapter. At the end of this thesis, remarks on shadowing in general multistep methods and remarks on a relationship between the shadowing property and stability are presented. 1
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Numerical Solution of Convection-dominated Problems
Lukáš, Petr ; Knobloch, Petr (advisor) ; Burman, Erik (referee) ; Linß, Torsten (referee)
Title: Numerical Solution of Convection-dominated Problems Author: Petr Lukáš Department: Department of Numerical Mathematics Supervisor: doc. Mgr. Petr Knobloch, Dr., DSc., Department of Numerical Mathematics Abstract: Numerical solution of the scalar convection-diffusion-reaction problems often possesses spurious oscillations which appear in the discrete solution when convection dominates diffusion and standard non-adaptive discretizations are used. Numerical solution of convection-dominated problems requires special techniques to suppress these oscillations. Often stabilized methods are applied which involve free parameters. These parameters significantly influence the quality of the solution but their optimal choice is usually not known. In this work we define them in an adaptive way by minimizing an error indicator characterizing the quality of the approximate solution. We consider new nonlinear limited-memory optimization methods. A nontrivial requirement on the error indicator is that its minimization with respect to the stabilization parameters should suppress spurious oscillations without smearing layers. In this thesis novel error indicators are introduced and their suitability is considered on different benchmarks. Keywords: FEM, optimization, SUPG, SDFEM, SOLD, Error indicator
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Numerical solution of convection-diffusion equations using stabilization and adaptive methods
Lamač, Jan ; Knobloch, Petr (advisor) ; Dolejší, Vít (referee)
The subject of the present Master Thesis is a comparison of numerical solution of convection-diffusion equations aproaches using stabilization and adaptive methods. Firstly the basic aspects and thoughts of employed numerical method - Galerkin finite element method - are summarized. Consequently the most common kinds of stabilization methods for spurious oscillations diminishing are defined (esp. SUPG method). Next section is devoted to a posteriori error estimations and adaptive refinement of triangulation which could help to diminish the spurious oscillations too. All mentioned methods and techniques are implemented and finally tested on the sample examples.
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Mixed finite element method for the Poisson equation
Švihlová, Helena ; Knobloch, Petr (advisor) ; Dolejší, Vít (referee)
The aim of this bachelor thesis is the implementation of the mixed element method for the Poisson equation and the comparison with results of the classical finite element method. The thesis is divided into two chapters. In the first chapter there are descriptions of the spaces occurring in the weak formulation of the Poisson equation and descriptions of the spaces which are suitable to approach them. The second chapter studies the existence of the solutions of the approximated tasks and their convergence. The main part of this thesis are schemes of the solutions of both methods and the tables comparing errors of these solutions for three diferent functions. 1
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Computer simulation of radiobiological effect of oxygen
Havle, Oto ; Felcman, Jiří (advisor) ; Knobloch, Petr (referee)
Exposing a live cell to ionizing radiation can lead to cell inactivation. Oxygen and other chemical substances soluted in the internal environment of a cell participate in various chemical reactions during the chemical stage of the radiobiological process. The thesis is concerned with modelling di usion and reaction processes of the chemical stage. A system of semilinear parabolic partial di erential equations for concentrations of chemical substances involved is derived in three spatial dimensions, including weak formulation. Assuming spherical symmetry, the problem can be reduced to one dimension. Under further simpli cations, existence of solution is proven via Galerkin method. The one-dimensional problem is solved numerically, using nite element discretization. Error estimates and computer implementation are presented.
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Fourier method for solving partial differential equations
Tůma, Karel ; Pokorný, Milan (advisor) ; Knobloch, Petr (referee)
Na./cv prace: Fouricrova metoda pro feseni parc.ialnich dirornncialnich rovnic Autor: Karri Tuma Katedra (ust.av): Matematicky ust.av UK Vedouci bakalafske praoo: Mgr. Milan Pokorny, Ph.D. e-mail vodouciho: pokorny@karlin.mff.cuni.cz Abstra.kt: V pfedlo/ene praci odvodime rovnici vedeni tepla a.rovnici slruny. Ty pak nasledno. fesime v jodno prost.orove dimenxi ponioci Fonricrovy me- tody apocivajfci v separaci promennych a nale/eni feseni vc l.varn ncko- nccnc' fady. Zaljyvainc so t.fonii ru/nymi okrajovynii podininkanii. Dah^ vy- sotrujomo vlastnosii foscni tcchlo dvou problcmu. Provadinio analyzu kon- vorgtuicc fx'soni vu tvaru fad v -/avislosti na pocat.ocnich podminkach uloh. Uka/c-me. /o pornoci Fouriorovy inolody l/.c fosil lako stucionarni ulohy, konkrctno so zabyvanio Laplaccoviju rovnici s okrajovynii podminkami na ruznych oblasloch (kruh. vyscc. vyscc mc/ikru/f, mraikru/i). Klicova slova: Parcialni diforoncialni rovnico, Fouricrova tnot.oda, rovnico vodoni lopla, rovnico sLruny. Title: Fourier method for solving partial differential equations Author: Karel Tuma, Department: Matematicky ustav UK Supervisor: Mgr. Milan Pokorny. Ph.D. Supervisor's e-mail address: pokorny@karlin.raff.cuni.cz Abstract: In the present work we derive the heat equation and the wave equation. They arc- solved in one space...
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Mathematical Analysis and Numerical Computation of Volume-Constrained Evolutionary Problems Involving Free Boundaries
Švadlenka, Karel ; Feistauer, Miloslav (advisor) ; Křížek, Michal (referee) ; Knobloch, Petr (referee)
The object of study of the present thesis are evolutionary problems satisfying volume preservation condition, i.e., problems whose solution have a constant value of the integral of their graph. In particular, the following types of problems with volume constraint are dealt with: parabolic problem (heat-type), hyperbolic problem (wave-type), parabolic free-boundary problem (heat-type with obstacle) and hyperbolic free-boundary problem (degenerate wave-type with obstacle). The key points are design of equations, proof of existence of weak solutions to them and development of numerical methods and algorithms for such problems. The main tool in both the theoretical analysis and the numerical computation is the discrete Morse flow, a variational method consisting in discretizing time and stating a minimization problem on each time-level. The volume constraint appears in the equation as a nonlocal nonlinear Lagrange multiplier but it can be handled elegantly in discrete Morse flow method by restraining the set of admissible functions for minimization. The theory is illustrated with results of numerical experiments.
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Computer simulation and numerical analysis of compressible flow problems
Kubera, Petr ; Felcman, Jiří (advisor) ; Knobloch, Petr (referee) ; Fürst, Jiří (referee)
The thesis deals with the construction of an adaptive 1D and 2D mesh in the framework of the cell- centered finite volume scheme. The adaptive strategy is applied to the numerical solution of problems governed by the Euler equations, which is a hyperbolic system of PDE's. The used algorithm is applicable to nonstationary problems and consists of three independent parts, which are cyclically repeated. These steps are PDE evolution, then mesh adaptation and recovery of numerical solution from the old mesh to the newly adapted mesh. Owing to this the algorithm can be used also for other hyperbolic systems. The thesis is focused on the development of our mesh adaptation strategy, based on the anisotropic mesh adaptation, which preserves the geometric mass conservation law in each computational step. The proposed method is suitable to solve problems with moving discontinuities. Several test problems with moving discontinuity are computed to compare our algorithm with Moving Mesh algorithms.
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Choice of the SUPG parameter for higher order finite elements
Kohutka, Jiří ; Knobloch, Petr (advisor) ; Dolejší, Vít (referee)
In this work, we deal with the finite element method Streamline Upwind/Petrov-Galerkin (SUPG) and use it to solve boundary value problem for the stationary convection-diffusion equation with dominant convection with Dirichlet boundary condition on the whole boundary of bounded polyhedral computational domain of dimension 1 and 2, respectively. We consider a quadratic Lagrangian finite elements on the line segments and triangles, respectively. The core of the work is a proposition of choice of stabilizing parameter of SUPG method as an elementwise affine function in outflow boundary layer and as an elementwise constant function in the rest of the computational domain. We show that this choice gives a more accurate solution than the choice of the stabilization parameter as a constant in each element. 1
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