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Higher-order approximations in the finite element methods
Kolman, Karel ; Křížek, Michal (advisor) ; Feistauer, Miloslav (referee) ; Mlýnek, Jaroslav (referee)
A survey of existing nite element superconvergence theory is conducted. The Steklov postprocessing operator is presented and its properties are investigated, superconvergence bounds are proved. A two-level method for eigenvalue problems is generalized for application to a nonselfadjoint problem. Applications of superconvergence theory to variationally formulated eigenvalue problems are discussed. Numerical experiments documenting the methods are performed.
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Solution of inverse problem for a flow around an airfoil
Šimák, Jan ; Feistauer, Miloslav (advisor) ; Felcman, Jiří (referee) ; Sváček, Petr (referee)
Title: Solution of inverse problem for a flow around an airfoil Author: Mgr. Jan Šimák Department: Department of Numerical Mathematics Supervisor: prof. RNDr. Miloslav Feistauer, DrSc., dr. h. c., Department of Numerical Mathematics Abstract: The method described in this thesis deals with a solution of an inverse problem for a flow around an airfoil. It can be used to design an airfoil shape according to a specified velocity or pressure distribution along the chord line. The method is based on searching for a fixed point of an operator, which combines an approximate inverse and direct operator. The approximate inverse operator, derived on the basis of the thin airfoil theory, assigns a corresponding shape to the specified distribution. The resulting shape is then constructed using the mean camber line and thickness function. The direct operator determines the pressure or velocity distribution on the airfoil surface. We can apply a fast, simplified model of potential flow solved using the Fredholm integral equation, or a slower but more accurate model of RANS equations with a k-omega turbulence model. The method is intended for a subsonic flow.
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Numerical analysis of problems in time-dependent domains
Balázsová, Monika ; Feistauer, Miloslav (advisor) ; Dumbser, Michael (referee) ; Bause, Markus (referee)
This work is concerned with the theoretical analysis of the space-time discontinuous Galerkin method applied to the numerical solution of nonstationary nonlinear convection-diffusion problem in a time- dependent domain. At first, the problem is reformulated by the use of the arbitrary Lagrangian-Eulerian (ALE) method, which replaces the classical partial time derivative by the so-called ALE derivative and an additional convection term. Then the problem is discretized with the use of the ALE space-time discontinuous Galerkin method. On the basis of a technical analysis we obtain an unconditional stability of this method. An important step in the analysis is the generalization of a discrete characteristic function associated with the approximate solutionin a time-dependentdomainand the derivationof its properties. Further we derive an a priori error estimate of the method in terms of the interpolation error, as well as in terms of h and tau. Finally, some practical applications of the ALE space-time discontinuos Galerkin method in a time-dependent domain are given. We are concerned with the numerical solution of a nonlinear elasticity benchmark problem and moreover with the interaction of compressible viscous flow with elastic structures. The main attention is paid to the modeling of flow induced vocal fold...
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Some aspects of the discontinuous Galerkin method for the solution of convection-diffusion problems
Balázsová, Monika ; Feistauer, Miloslav (advisor)
In the present work we deal with the stability of the space-time discontinuous Galerkin method applied to non-stationary, 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 non-symmetric, 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.
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Numerical solution of compressible flow
Prokopová, Jaroslava ; Feistauer, Miloslav (advisor)
This work deals with the problem of inviscid, compressible flow in a timedependent domain. We describe mathematical properties of the Euler equations and the system of governing equations is solved with the aid of the discontinuous Galerkin finite element method (DGFEM) in the time-indepentent domain. The main aim of this work is the study of this problem in time-dependent domains. For this reason the Arbitrary Lagrangian-Eulerian (ALE) method is presented. The governing equations are formulated in the ALE formulation and discretized in space and time by the DGFEM. Shortly we mention the shock capturing of the obtained scheme and the solution of the resulting linear system with the aid of Generalized Minimal Residual (GMRES) method. At the end of this work we present and compare results obtained by two different ALE formulations of the governing equations in the rectangular domain with a moving part of lower wall.
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Spline-base functions for the soulution of boundary-value problems
Horčička, Martin ; Dolejší, Vít (advisor) ; Feistauer, Miloslav (referee)
Solving the Poisson equation using finite element method with a basis com- posed of natural cubic splines. In this thesis we introduce the notion of weak derivatives, Sobolev spaces and formulate the weak form of the Poisson equation in order to build up to the finite element method. Furthermore, the thesis contains a construction of a natural cubic spline and a description of the used basis. The computed solution approximates well the exact solution, especially if the right side satisfies certain conditions. 1
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HP-FEM for Coupled Problems in Fluid Dynamics
Dubcová, Lenka ; Feistauer, Miloslav (advisor) ; Segeth, Karel (referee) ; Dolejší, Vít (referee)
The thesis is concerned with the solution of multiphysics problems described by partial differential equations using higher-order finite element method (hp-FEM). Basics of hp-FEM are described, together with some practical details and challenges. The hp-adaptive strategy, based on the reference solution and meshes with arbitrary level hanging nodes, is discussed. The thesis is mainly concerned with the extension of this strategy to monolithical solution of coupled multiphysics problems, where each physical field exhibits different qualitative behavior. In such problems, each physical field is discretized on an individual mesh automatically obtained by the adaptive algorithm to suit the best the corresponding solution component. Moreover, the meshes can change in time, following the needs of the solution components. All described methods and technologies are demonstrated on several examples throughout the thesis, where comparisons with traditionally used approaches are shown.
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Some aspects of the discontinuous Galerkin method for the solution of convection-diffusion problems
Balázsová, Monika ; Feistauer, Miloslav (advisor)
In the present work we deal with the stability of the space-time discontinuous Galerkin method applied to non-stationary, 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 non-symmetric, 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.
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Discontinuous Galerkin method for the solution of boundary-value problems in non-smooth domains
Bartoš, Ondřej ; Feistauer, Miloslav (advisor) ; Dolejší, Vít (referee)
This thesis is concerned with the analysis of the finite element method and the discontinuous Galerkin method for the numerical solution of an elliptic boundary value problem with a nonlinear Newton boundary condition in a two-dimensional polygonal domain. The weak solution loses regularity in a neighbourhood of boundary singularities, which may be at corners or at roots of the weak solution on edges. The main attention is paid to the study of error estimates. It turns out that the order of convergence is not dampened by the nonlinearity, if the weak solution is nonzero on a large part of the boundary. If the weak solution is zero on the whole boundary, the nonlinearity only slows down the convergence of the function values but not the convergence of the gradient. The same analysis is carried out for approximate solutions obtained with numerical integration. The theoretical results are verified by numerical experiments. 1
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