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Superconvergence for discontinuous Galerkin time discretizations
Roskovec, Filip ; Vlasák, Miloslav (advisor) ; Knobloch, Petr (referee)
The topic of this thesis is the application of the discontinuous Galerkin finite element method (DGFEM) on space-time discretizations of simple nonstationary problems. Unlike the standard finite element method, discontinuous Galerkin method does not require any continuity between neighbouring elements. We apply the DGFEM separately in space and in time. At first, we implement discretization with respect to space variables, whereby we acquire the space semidiscretization. Subsequently we apply Time discontinuous Galerkin method to the problem. We seek the aproximate solution in the space of discontinuous piecewise polynomial functions of degree p in space and degree q in time. This is followed by the error estimates of this scheme. In the end we examine the supercovergence behaviour of the scheme in nodes of the time discretization. The theoretical results are verified by numerical experiments.
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Low-rank matrix approximations
Jarolímová, Alena ; Tůma, Miroslav (advisor) ; Vlasák, Miloslav (referee)
This thesis is focused on using low rank matrices in numerical mathematics. We introduce conjugate gradient method and its preconditioning which we use in other chapters. Then we describe four different approaches to approximation using low rank matrices. First we discuss classical approximation using singu- lar value decomposition. Next, using a model problem, we describe hierarchical matrices, which are connected with applications in physics and technique. Then pseudo-skeleton decomposition is introduced. We formulate and prove a theorem about error estimate of this decomposition. We also mention algorithm Maxvol which can compute pseudo-skeletal decomposition of tall matrices. Next chapter is dedicated to probabilistic algorithms and to least-squares solver Blendenpik. In conclusions we show results of experiments focused on preconditioning using algorithm Maxvol. 1
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Line search in descent methods
Moravová, Adéla ; Tichý, Petr (advisor) ; Vlasák, Miloslav (referee)
In this thesis, we deal with descent methods for functional minimalization. We discuss three conditions for the choice of the step length (Armijo, Goldstein, and Wolfe condition) and four descent methods (The steepest descent method, Newton's method, Quasi-Newton's method BFGS and the conjugate gradient method). We discuss their convergence properties and their advantages and dis- advantages. Finally, we test these methods numerically in the GNU Octave pro- gramming system on three different functions with different number of variables. 1
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Low-rank matrix approximations
Jarolímová, Alena ; Tůma, Miroslav (advisor) ; Vlasák, Miloslav (referee)
This thesis is focused on using low rank matrices in numerical mathematics. We introduce conjugate gradient method and its preconditioning which we use in other chapters. Then we describe four different approaches to approximation using low rank matrices. First we discuss classical approximation using singu- lar value decomposition. Next, using a model problem, we describe hierarchical matrices, which are connected with applications in physics and technique. Then pseudo-skeleton decomposition is introduced. We formulate and prove a theorem about error estimate of this decomposition. We also mention algorithm Maxvol which can compute pseudo-skeletal decomposition of tall matrices. Next chapter is dedicated to probabilistic algorithms and to least-squares solver Blendenpik. In conclusions we show results of experiments focused on preconditioning using algorithm Maxvol. 1
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Numerical solution of convection-diffusion problems by discontinuous Galerkin method
Vlasák, Miloslav ; Dolejší, Vít (advisor) ; Janovský, Vladimír (referee) ; Vejchodský, Tomáš (referee)
This work is concerned with the theoretical analysis of the discontinuous Galerkin finite element method. We use a discontinuous Galerkin formulation for a scalar convection-diffusion equation with nonlinear convective term. The resulting semidiscretized equations with symmetric (SIPG) or nonsymmetric (NIPG) diffusive term are then discretized in time by Backward Differential formulae (BDF), implicit Runge-Kutta methods and Time discontinuous Galerkin. All of these schemes are linearized by a suitable explicit extrapolations to avoid nonlinearity in the convective term. These final schemes are theoretically analyzed and error estimates are derived. We also present some superconvergence result for Time discontinuous Galerkin for nonsymmetric operator. Numerical experiments verify the theoretical results.
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Numerical solution of nonlinear transport problems
Bezchlebová, Eva ; Feistauer, Miloslav (advisor) ; Vlasák, Miloslav (referee)
Práce je zaměřená na numerickou simulaci dvoufázového proudění. Je studován matematický model a numerická aproximace toku dvou nemísitelných nestlačitelných tekutin. Rozhraní mezi tekutinami je popsáno pomocí pomocí tzv. level set metody. Představena je diskretizace problému v prostoru a v čase. Metoda konečných prvk· se zpětnou Eulerovou metodou je aplikována na Navierovy-Stokesovy rovnice a časoprostorová nespojitá Galerkinova metoda je použita k řešení transportního problému. D·raz je kladen na analýzu chyby nespojité Galerkinovy metody přímek a časoprostorové nespojité Galerkinovy metody pro transportní problém. Jsou prezentovány numerické výsledky. 1
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