Title: Estimation of the algebraic error and stopping criteria in numerical solution of partial differential equations Author: Jan Papež Department: Department of Numerical Mathematics Supervisor of the master thesis: Zdeněk Strakoš Abstract: After introduction of the model problem and its properties we describe the Conjugate Gradient Method (CG). We present the estimates of the energy norm of the error and a heuristic for the adaptive refinement of the estimate. The difference in the local behaviour of the discretization and the algebraic error is illustrated by numerical experiments using the given model problem. A posteriori estimates for the discretization and the total error that take into account the inexact solution of the algebraic system are then discussed. In order to get a useful perspective, we briefly recall the multigrid method. Then the Cascadic Conjugate Gradient Method of Deuflhard (CCG) is presented. Using the estimates for the error presented in the preceding parts of the thesis, the new stopping criteria for CCG are proposed. The CCG method with the new stopping criteria is then tested. Keywords: numerical PDE, discretization error, algebraic error, error es- timates, locality of the error, adaptivity

In the beginning we outline some important properties of blood and de- scribe it from the biological point of view. In the next section we show how we derived our model based on the mixture theory. For the final model we suggest a mathematical method based on the finite element method and subject it to tests for flow in a simple domain. In the middle part we prove the existence of solution for a model with simplified constitutive relation for the stress tensor, which still includes an anisotropic model for the platelet diffusion. In the last section we show numerical results. We start with sim- ple testing computations in simple domains, followed by computations in a two-dimensional simulation of an aneurysm, and narrowed blood vessel re- spectively. In the end we also show some illustrative computations in three dimensions. 1

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In the work we modify the well-known minimal surface problem to a very special form, where the exponent two is replaced by a general positive parameter. To the modified problem we define four notions of solution in nonreflexive Sobolev space and in the space of functions of bounded variation. We examine the relationships between these notions to show that some of them are equivalent and some are weaker. After that we look for assumptions needed to prove the existence of solution to the problem in the sense of definitions provided. We outline that in the setting of spaces of functions of bounded variation the solution exists for any positive finite parameter and that if we accept some restrictions on the parameter then the solution exists in the Sobolev space, too. We also provide counterexample indicating that if the domain is non-convex, the solution in Sobolev space need not exist. Powered by TCPDF (www.tcpdf.org)

The aim of this work is to provide new results of global existence for dif- ferent evolution equations of fluid mechanics. We are in general interested in finding weak solutions without restrictions on the size of initial data. The proofs of existence are based on several different approaches including en- ergy methods, convergence analysis of finite numerical methods and convex integration. All these techniques significantly exploit results of mathematical analysis and other branches of mathematics. 1

In the thesis, results of a research focused on monitoring strategies used by students during problem solving are presented. It follows previous studies conducted with the use of think- aloud method. Due to limitations of this method, an eye-camera is used. The record of the camera is used not only to analyse student's problem-solving process, but also as a tool eliminating one of the RTA limits - inaccuracies due to forgetting. The strategies were investigated among economics-oriented vocational school students in the first grade. They solved indicator tasks from the Methodical Comments and Tasks for Educational Standards in Lower-Secondary Education. The research sample was selected based on a pre-test composed of the selected tasks. According to the results, the students were divided into four categories, two students of each category were selected for the research. They solved a second set of the tasks. Based on eye-tracking and think-aloud data, their problem- solving processes were mapped and false-positive results were eliminated. The research results show that the students used the most expansive strategies, mainly work with the periodical table. However, in most cases, these strategies did not lead to a correct solution, due to the students' problems during solving. In these cases, they...

The main aim of the master thesis titled "Syria and the concept responsibility to assist: new approach towards conflict resolution?" is to find out whether the extension of the doctrine of responsibility to protect by the concept of responsibility to assist that is based on the support of nonviolent movements could work as a new approach towards conflict resolution and thanks to this it could constitute another way of international assistance and civilian protection in cases when it is the regime that perpetrates the violence against its own population. For this reason the thesis, via the Syrian civil war that offers wide range of internal responses to the violence perpetrated by the regime against civilians, focuses on three different forms of international involvement and assistance. These are the international military intervention under the third pillar of R2P, the backing of domestic violent groups and insurgents and last but not least the assistance to nonviolent movements and civil initiatives in the form of R2A as a potential expansion of the second pillar under the R2P doctrine. These approaches are analyzed with regard to the issues of state sovereignty and moral responsibility to protect civilians that arises from the inconsistency of these two aspects in relation to any kind of...

We discuss certain aspects of string field theory and its applications in exploring the land- scape of classical string theory vacua. We start by giving a brief overview of various tree-level string field theories, as well as of some relevant mathematical background. As a byproduct of our general discussion of observables, we present a new gauge-invariant quantity for the A∞ formulation of open superstring field theory. Putting particular emphasis on perturba- tive methods, we proceed to review in detail the construction of tree-level effective actions governing the dynamics of a certain subset of degrees of freedom. In light of recent devel- opments, we also discuss efficient methods for evaluating certain vertices of zero-momentum effective actions for open superstring and heterotic string field theories in the presence of a global N = 2 worldsheet superconformal symmetry. We show how to apply this perturbative approach to study dynamics of the D(−1)/D3 system (both with and without a B-field), while also discussing a number of more complicated Dp-brane configurations. At generic points in their moduli spaces, such bound states of Dp-branes clearly cannot be described in terms of simple Dirichlet or Neumann boundary conditions. The rest of this thesis is therefore devoted to developing analytic...

The aim of submitted thesis is to bring a selection of journal texts by Růžena Jesenská, focused on period 1910-1930. This thesis includes, next to the edition of selected texts, edition note and commentary. The thesis reflects historical context and context of Růžena Jesenská's artworks, relates selected texts to significant artistic texts by Růžena Jesenská (Mimo svět, Z nepřístupných zahrad etc.)

In the present thesis we study the compressible Navier-Stokes-Fourier sys- tem. This is a system of partial differential equations describing the evolutionary problem for an adiabatic flow of a heat conducting compressible viscous fluid in a bounded domain. Here we consider the problem in two dimensions with zero Dirichlet boundary conditions for velocity. The cold pressure term in the pressure law for the momentum equation is here considered in the form pC(ϱ) ∼ ϱ logα (1+ϱ) for some α > 0, for which we need to work on the scale of Orlicz spaces in order to obtain useful estimates and in those space we formulate the problem weakly and also establish the weak compactness of the solution. The main result of this thesis is Theorem 6.1 where we show the existence of a weak solution with no assumptions on the size of the data and on arbitrary large time intervals. 1