
Some aspects of the discontinuous Galerkin method for the solution of convectiondiffusion problems
Balázsová, Monika ; Feistauer, Miloslav (advisor)
In the present work we deal with the stability of the spacetime discontinuous Galerkin method applied to nonstationary, nonlinear convection  diffusion problems. Discontinuous Galerkin method is a very efficient tool for numerical solution of partial differential equations, combines the advantages of the finite element method (polynomial approximations of high order of accuracy) and the finite volume method (discontinuous approximations). After the formulation of the continuous problem its discretization in space and time is described. In the formulation of the discontinuous Galerkin method the nonsymmetric, symmetric and incomplete version of discretization of the diffusion term is used and there are added penalty terms to the scheme also. In the third chapter are estimated individual terms of the previously derived approximate solution by special norms. Using the concept of discrete characteristic functions and the discrete Gronwall lemma, it is shown that the analyzed scheme is unconditionally stable. At the end, in the fourth chapter, are given some numerical experiments, which verify theoretical results from the previous chapter.


Incompressible and compressible viscous flow with low Mach numbers
Balázsová, M. ; Feistauer, M. ; Sváček, Petr ; Horáček, Jaromír
In this paper we compare incompressible flow and low Mach number compressible viscous flow. Incompressible NavierStokes equations were treated with the aid of discontinuous Galerkin method in space and backward difference method in time. We present numerical results for a flow in a channel which represents a simplified model of the human vocal tract. Presented numerical results give a good correspondence between the incompressible flow and the compressible flow with low Mach numbers.


Některé aspekty nespojité Galerkinovy metody pro řešení konvektivnědifuzních problémů
Balázsová, Monika ; Feistauer, Miloslav (advisor) ; Najzar, Karel (referee)
In the present work we deal with the stability of the spacetime discontinuous Galerkin method applied to nonstationary, nonlinear convection  diffusion problems. Discontinuous Galerkin method is a very efficient tool for numerical solution of partial differential equations, combines the advantages of the finite element method (polynomial approximations of high order of accuracy) and the finite volume method (discontinuous approximations). After the formulation of the continuous problem its discretization in space and time is described. In the formulation of the discontinuous Galerkin method the nonsymmetric, symmetric and incomplete version of discretization of the diffusion term is used and there are added penalty terms to the scheme also. In the third chapter are estimated individual terms of the previously derived approximate solution by special norms. Using the concept of discrete characteristic functions and the discrete Gronwall lemma, it is shown that the analyzed scheme is unconditionally stable. At the end, in the fourth chapter, are given some numerical experiments, which verify theoretical results from the previous chapter.


Nonsmooth Newton's method
Balázsová, Monika ; Haslinger, Jaroslav (advisor) ; Ligurský, Tomáš (referee)
In this thesis we generalize classical Newton's method for nonsmooth equations. For this purpose we define the Newton approximation of functions. Then we introduce several methods for solving equations with locally Lipschitz and piecewise smooth functions. We prove that their local convergence rate is Qsuperlinear or even Qquadratic. At the end we apply one of the algorithms to the beam problem with the obstacle. Based on the physical model we establish mathematical model and its discretization. Finally we implement the problem in the MATLAB. Results are summarized in tables.
