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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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Adaptive methods for singularly perturbed partial differential equations
Lamač, Jan ; Knobloch, Petr (advisor)
This thesis deals with solving singularly perturbed convection- diffusion equations. Firstly, we construct a matched asymptotic expansion of the solution of the singularly perturbed convection-diffusion equation in 1D and derive a formula for the zeroth-order asymptotic expansion in several two- dimensional polygonal domains. Further, we present a set of stabilization meth- ods for solving singularly perturbed problems and prove the uniform convergence of the Il'in-Allen-Southwell scheme in 1D. Finally, we introduce a modification of the streamline upwind Petrov/Galerkin (SUPG) method on convection-oriented meshes. This new method enjoys several profitable properties such as the ful- filment of the discrete maximum principle. Besides the analysis of the method and derivation of a priori error estimates in respective energy norms we also carry out several numerical experiments verifying the theoretical results.
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Adaptive methods for singularly perturbed partial differential equations
Lamač, Jan ; Knobloch, Petr (advisor)
This thesis deals with solving singularly perturbed convection- diffusion equations. Firstly, we construct a matched asymptotic expansion of the solution of the singularly perturbed convection-diffusion equation in 1D and derive a formula for the zeroth-order asymptotic expansion in several two- dimensional polygonal domains. Further, we present a set of stabilization meth- ods for solving singularly perturbed problems and prove the uniform convergence of the Il'in-Allen-Southwell scheme in 1D. Finally, we introduce a modification of the streamline upwind Petrov/Galerkin (SUPG) method on convection-oriented meshes. This new method enjoys several profitable properties such as the ful- filment of the discrete maximum principle. Besides the analysis of the method and derivation of a priori error estimates in respective energy norms we also carry out several numerical experiments verifying the theoretical results.
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Adaptive methods for singularly perturbed partial differential equations
Lamač, Jan ; Knobloch, Petr (advisor) ; Franz, Sebastian (referee) ; Vejchodský, Tomáš (referee)
This thesis deals with solving singularly perturbed convection- diffusion equations. Firstly, we construct a matched asymptotic expansion of the solution of the singularly perturbed convection-diffusion equation in 1D and derive a formula for the zeroth-order asymptotic expansion in several two- dimensional polygonal domains. Further, we present a set of stabilization meth- ods for solving singularly perturbed problems and prove the uniform convergence of the Il'in-Allen-Southwell scheme in 1D. Finally, we introduce a modification of the streamline upwind Petrov/Galerkin (SUPG) method on convection-oriented meshes. This new method enjoys several profitable properties such as the ful- filment of the discrete maximum principle. Besides the analysis of the method and derivation of a priori error estimates in respective energy norms we also carry out several numerical experiments verifying the theoretical results.
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Numerical solution of convection-diffusion equations using stabilization and adaptive methods
Lamač, Jan ; Dolejší, Vít (referee) ; Knobloch, Petr (advisor)
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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