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Multilevel methods
Vacek, Petr ; Strakoš, Zdeněk (advisor)
The analysis of the convergence behavior of the multilevel methods is in the literature typically carried out under the assumption that the problem on the coarsest level is solved exactly. The aim of this thesis is to present a description of the multilevel methods which allows inexact solve on the coarsest level and to revisit selected results presented in literature using these weaker assumptions. In particular, we focus on the derivation of the uniform bound on the rate of convergence. Moreover, we discuss the possible dependence of the convergence behavior on the mesh size of the initial triangulation. 41
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Multilevel methods
Vacek, Petr ; Strakoš, Zdeněk (advisor) ; Pultarová, Ivana (referee)
The analysis of the convergence behavior of the multilevel methods is in the literature typically carried out under the assumption that the problem on the coarsest level is solved exactly. The aim of this thesis is to present a description of the multilevel methods which allows inexact solve on the coarsest level and to revisit selected results presented in literature using these weaker assumptions. In particular, we focus on the derivation of the uniform bound on the rate of convergence. Moreover, we discuss the possible dependence of the convergence behavior on the mesh size of the initial triangulation. 41
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Multilevel incomplete factorizations
Mudroňová, Veronika ; Tůma, Miroslav (advisor) ; Strakoš, Zdeněk (referee)
In this work we present incomplete matrix decomposition algorithms including their multi-level extensions. First, we have mentioned classical standard strategies that are used for the solution of linear systems. Second, we present incomplete methods based on the hierarchical approach known as multilevel matrix decom- positions. Just this latter type of approaches is considered for numerical expe- riments. Namely, two major hierarchical algorithms, the greedy search with the strongest coupling and the greedy search with the smallest degree, are compared, using the Fortran programming language. The results presented in the Matlab figures are commented and discussed. 1
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Multilevel methods and adaptivity
Vacek, Petr ; Strakoš, Zdeněk (advisor) ; Tichý, Petr (referee)
After introduction of the model problem we derive its weak formulation, show the existence and the uniqueness of the solution, and present the Galerkin finite element method. Then we briefly describe some of the stationary iterative methods and their smoothing property. We present the most common multigrid schemes, i.e. two-grid correction scheme, V-cycle scheme, and the full multigrid algorithm. Then we perform numerical experiment showing the differences between the use of the direct and iterative coarsest grid solver in V-cycle scheme and experiment considering a perturbation of the correction vector simulating a fault of a computational device. Powered by TCPDF (www.tcpdf.org)
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