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A posteriori error estimates for numerical solution of convection-difusion problems
Šebestová, Ivana ; Dolejší, Vít (advisor) ; Sváček, Petr (referee) ; Brandts, Jan (referee)
This thesis is concerned with several issues of a posteriori error estimates for linear problems. In its first part error estimates for the heat conduction equation discretized by the backward Euler method in time and discontinuous Galerkin method in space are derived. In the second part guaranteed and locally efficient error estimates involving algebraic error for Poisson equation discretized by the discontinuous Galerkin method are derived. The technique is based on the flux reconstruction where meshes with hanging nodes and variable polynomial degree are allowed. An adaptive strategy combining both adaptive mesh refinement and stopping criteria for iterative algebraic solvers is proposed. In the last part a numerical method for computing guaranteed lower and upper bounds of principal eigenvalues of symmetric linear elliptic differential operators is presented. 1
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The choice of the stopping criteria for Newton-like methods
Kurnas, Jakub ; Dolejší, Vít (advisor) ; Tichý, Petr (referee)
We formulate examples of partial differential equations which can be solved through their discretization and subsequent solution of derived algebraic system. A brief summary of Discontinuous Galerkin Discretization is given as well as definitions of algebraic and discretization errors. We derive the Newton method, which solves nonlinear algebraic systems by solving a sequence of linear problems, we modify the method and examine implementation options. We define stopping criteria for the Newton-like method using aforementioned errors and we explain how to keep accuracy of the solution of derived algebraic system and the original partial differential equation in balance. We present numerical experiments to illustrate theoretical background and mention several basic properties of the Newton- like method.
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A posteriori error estimates for numerical solution of convection-difusion problems
Šebestová, Ivana ; Dolejší, Vít (advisor) ; Sváček, Petr (referee) ; Brandts, Jan (referee)
This thesis is concerned with several issues of a posteriori error estimates for linear problems. In its first part error estimates for the heat conduction equation discretized by the backward Euler method in time and discontinuous Galerkin method in space are derived. In the second part guaranteed and locally efficient error estimates involving algebraic error for Poisson equation discretized by the discontinuous Galerkin method are derived. The technique is based on the flux reconstruction where meshes with hanging nodes and variable polynomial degree are allowed. An adaptive strategy combining both adaptive mesh refinement and stopping criteria for iterative algebraic solvers is proposed. In the last part a numerical method for computing guaranteed lower and upper bounds of principal eigenvalues of symmetric linear elliptic differential operators is presented. 1
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Algebraic Error in Matrix Computations in the Context of Numerical Solution of Partial Differential Equations
Papež, Jan ; Strakoš, Zdeněk (advisor) ; Ramage, Alison (referee) ; Vejchodský, Tomáš (referee)
Title: Algebraic Error in Matrix Computations in the Context of Numerical Solution of Partial Differential Equations Author: Jan Papež Department: Department of Numerical Mathematics Supervisor: prof. Ing. Zdeněk Strakoš, DrSc., Department of Numerical Mathe- matics Abstract: Solution of algebraic problems is an inseparable and usually the most time-consuming part of numerical solution of PDEs. Algebraic computations are, in general, not exact, and in many cases it is even principally desirable not to perform them to a high accuracy. This has consequences that have to be taken into account in numerical analysis. This thesis investigates in this line some closely related issues. It focuses, in particular, on spatial distribution of the errors of different origin across the solution domain, backward error interpretation of the algebraic error in the context of function approximations, incorporation of algebraic errors to a posteriori error analysis, influence of algebraic errors to adaptivity, and construction of stopping criteria for (preconditioned) iterative algebraic solvers. Progress in these issues requires, in our opinion, understanding the interconnections between the phases of the overall solution process, such as discretization and algebraic computations. Keywords: Numerical solution of partial...
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