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A posteriori error estimates for numerical solution of convection-difusion problems
Šebestová, Ivana ; Dolejší, Vít (advisor) ; Sváček, Petr (referee) ; Brandts, Jan (referee)
This thesis is concerned with several issues of a posteriori error estimates for linear problems. In its first part error estimates for the heat conduction equation discretized by the backward Euler method in time and discontinuous Galerkin method in space are derived. In the second part guaranteed and locally efficient error estimates involving algebraic error for Poisson equation discretized by the discontinuous Galerkin method are derived. The technique is based on the flux reconstruction where meshes with hanging nodes and variable polynomial degree are allowed. An adaptive strategy combining both adaptive mesh refinement and stopping criteria for iterative algebraic solvers is proposed. In the last part a numerical method for computing guaranteed lower and upper bounds of principal eigenvalues of symmetric linear elliptic differential operators is presented. 1
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Rovnice vedení tepla ve fyzice planetek a meteoroidů
Pohl, Leoš ; Brož, Miroslav (advisor) ; Vokrouhlický, David (referee)
Non-gravitational forces caused by thermal emission of photons can significantly change orbits and spin states of asteroids in the long term. A solution of the Heat Conduction Equation (HCE) in an asteroid is necessary to evaluate the forces. Finite Difference Methods (FDMs) are implemented in a Fortran numerical HCE solver to calculate a temperature distribution within a system of 1-dimensional slabs which approximate the asteroid. We compare the methods w.r.t. convergence, accuracy and computational efficiency. The numerical results are compared with a simplified steady-state analytical solution. We calculate the non-gravitational accelerations and resulting semimajor axis drift from the numerical results. The implemented FDMs are shown to be convergent with denser grids and the best method has been selected. The analytical solution provides a good first-guess estimate of the temperature amplitude. The drift in semimajor axis of the tested asteroids, which is due to the non-gravitational forces, is in order-of-magnitude agreement with more accurate models and observational data.
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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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Heat diffusion equation and thermophysical modelling of asteroids
Pohl, Leoš ; Ďurech, Josef (advisor) ; Čapek, David (referee)
Light curve inversion is a standard method to determine shapes, rotation periods and spin axis orientations of asteroids. This method can be extended to determine the size, albedo, thermal inertia and surface roughness parameters of an asteroid by including observations in thermal infrared. A solution of the Heat Conduction Equation (HCE) is necessary to model infrared flux from the asteroid. We analyse the accuracy requirements of the extended method for numerical solution of the HCE. We show that current implementation leads to errors in flux that are substantial. We recommend changes in the current implementation of the HCE solving approach to address the accuracy issues. We discuss uniqueness and stability of the solutions produced by the extended method as well as the accuracy of the determined parameters and their stability. Shapes of asteroids are produced and their physical attributes are determined based on light curve and infrared data.
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A posteriori error estimates for numerical solution of convection-difusion problems
Šebestová, Ivana ; Dolejší, Vít (advisor) ; Sváček, Petr (referee) ; Brandts, Jan (referee)
This thesis is concerned with several issues of a posteriori error estimates for linear problems. In its first part error estimates for the heat conduction equation discretized by the backward Euler method in time and discontinuous Galerkin method in space are derived. In the second part guaranteed and locally efficient error estimates involving algebraic error for Poisson equation discretized by the discontinuous Galerkin method are derived. The technique is based on the flux reconstruction where meshes with hanging nodes and variable polynomial degree are allowed. An adaptive strategy combining both adaptive mesh refinement and stopping criteria for iterative algebraic solvers is proposed. In the last part a numerical method for computing guaranteed lower and upper bounds of principal eigenvalues of symmetric linear elliptic differential operators is presented. 1
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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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Rovnice vedení tepla ve fyzice planetek a meteoroidů
Pohl, Leoš ; Brož, Miroslav (advisor) ; Vokrouhlický, David (referee)
Non-gravitational forces caused by thermal emission of photons can significantly change orbits and spin states of asteroids in the long term. A solution of the Heat Conduction Equation (HCE) in an asteroid is necessary to evaluate the forces. Finite Difference Methods (FDMs) are implemented in a Fortran numerical HCE solver to calculate a temperature distribution within a system of 1-dimensional slabs which approximate the asteroid. We compare the methods w.r.t. convergence, accuracy and computational efficiency. The numerical results are compared with a simplified steady-state analytical solution. We calculate the non-gravitational accelerations and resulting semimajor axis drift from the numerical results. The implemented FDMs are shown to be convergent with denser grids and the best method has been selected. The analytical solution provides a good first-guess estimate of the temperature amplitude. The drift in semimajor axis of the tested asteroids, which is due to the non-gravitational forces, is in order-of-magnitude agreement with more accurate models and observational data.
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