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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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Estimations of the remainders of the Lagrange interpolation formula and Newton - Cotes quadrature
Bezchlebová, Eva ; Kofroň, Josef (advisor) ; Najzar, Karel (referee)
Hlavním tématem práce je zkoumání Newton-Cotesovy kvadratury. V první řadě se budeme zabývat Lagrangeovskou interpolační formulí, ze které zmiňovaná kvadratura vychází. Zde bu- deme klást d·raz na alternativní odhady zbytku této interpolace a její m-té derivace, které nejsou příliš známé. Cílem je odhady uspořádat a provést d·kladné d·kazy získaných výsledk·. Dále se pokusíme najít optimální kvadraturní formuli ve smyslu nejmenšího odhadu chyby, což se budeme snažit ukázat na příkladech, kdy porovnáme námi získanou formuli s jinými známými kvadraturami. V neposlední řadě se zaměříme na samotnou Newton-Cotesovu kvadraturu a to především z hlediska její konvergence, resp. divergence. Uvedené závěry o konvergenci, resp. di- vergenci této kvadratury poté stvrdíme numerickými experimenty, ukazující chování kvadratury pro r·zné třídy funkcí. 1
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