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Automatic hp-adaptivity on Meshes with Arbitrary-Level Hanging Nodes in 3D
Kůs, Pavel
The thesis is concerned with theoretical and practical aspects of the hp- adaptive finite element method for solving elliptic and electromagnetic prob- lems described by partial differential equations in three spatial dimensions. Besides the standard element refinements, the hp-adaptivity allows indepen- dent adaptation of degrees of the polynomial approximation as well. This leads to exponentially fast convergence even for problems with singularities. The efficiency of the hp-adaptivity is enhanced even more by the ability of the algorithm to work with meshes with arbitrary-level hanging nodes. This generality, however, leads to great complexity of the implementation. There- fore, the thesis concentrates on the mathematical analysis of algorithms that have led to successful implementation of the method. In addition, the the- sis discusses the numerical integration in 3D and the implementation of the method itself. Finally, numerical results obtained by this new implemen- tation are presented. They confirm advantages of hp-adaptivity on meshes with arbitrary-level hanging nodes. 1
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Automatic hp-adaptivity on Meshes with Arbitrary-Level Hanging Nodes in 3D
Kůs, Pavel ; Vejchodský, Tomáš (advisor) ; Segeth, Karel (referee) ; Dolejší, Vít (referee)
The thesis is concerned with theoretical and practical aspects of the hp- adaptive finite element method for solving elliptic and electromagnetic prob- lems described by partial differential equations in three spatial dimensions. Besides the standard element refinements, the hp-adaptivity allows indepen- dent adaptation of degrees of the polynomial approximation as well. This leads to exponentially fast convergence even for problems with singularities. The efficiency of the hp-adaptivity is enhanced even more by the ability of the algorithm to work with meshes with arbitrary-level hanging nodes. This generality, however, leads to great complexity of the implementation. There- fore, the thesis concentrates on the mathematical analysis of algorithms that have led to successful implementation of the method. In addition, the the- sis discusses the numerical integration in 3D and the implementation of the method itself. Finally, numerical results obtained by this new implemen- tation are presented. They confirm advantages of hp-adaptivity on meshes with arbitrary-level hanging nodes. 1
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Numerical solution of convection-diffusion equations with the aid of adaptive time-space higher order methods
Kůs, Pavel ; Felcman, Jiří (referee) ; Dolejší, Vít (advisor)
This thesis deals with solution of scalar nonlinear convection-diffusion equation with aid of discontinuous Galerkin method. It's aim is to implement an adaptive choice of time step. To do this, we derived 2 sufficiently stable methods for solution of systems of ordinary differential equations obtained by space semidicretization, which is carried out by the discontinuous Galerkin method. Using those two approximate solutions, we estimate local error of discretization. Using it, we are able to choose following time step in such way, that local error is approximately equal to given tolerance. Several numerical simulations were carried out to check properties of this method.
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Integration in higher-order finite element method in 3D
Kůs, Pavel
Integration of higher-order basis functions is an important issue, that is not as straightforward as it may seem. In traditional low-order FEM codes, the bulk of computational time is a solution of resulting system of linear equations. In the case of higher-order elements the situation is different. Especially in three dimensions the time of integration may represent significant part of the computation.
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Řešení 3D elektrostatických problémů se singulaturou s použitím adaptivní hp-FEM
Kůs, Pavel ; Šolín, Pavel ; Doležel, Ivo
For most numerical methods, accurate resolution of singularities occurring at sharp re-entrant corners or edges of electrically conductive objects is highly problematic. Finite differences are known for their inability to treat complex geometries, and traditional low-order (piecewise-linear or quadratic) finite element methods (FEM) exhibit extremely poor convergence. Nowadays, the best numerical method for the solution of most singular problems is the adaptive hp-version of the FEM (hp-FEM). This method is based on spatial refinements toward the singularities combined with optimal variation of polynomial degrees on the elements. The hp-FEM has mathematically proven exponential convergence, and also in practical computations typically it is by several orders of magnitudes faster than standard FEM.
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