Dissertation ueber die effektive Vorkonditionierung linearer Gleichungssysteme resultierend aus der Diskretisierung eines elliptischen Randwertproblems 2. Ordnung mittels der Methode der Finiten Elementen. Als Vorkonditionierer werden multi-level artige Vorkonditionierer (BPX, Multi-grid, Wavelets) benutzt.

In dieser Arbeit werden stochastische Charakteristiken der Lösung parabolischer Differentialgleichungen mit zufälligen Neumann-Randbedingungen mit Hilfe der Finite-Elemente-Methode angegeben. Dabei wird der Berechnung der Korrelations- bzw. Varianzfunktion besondere Bedeutung beigemessen. Das stochastische Randanfangswertproblem wird durch Anwendung von FEM-Techniken durch ein System gewöhnlicher Differentialgleichungen mit stochastischen inhomogenen Termen approximiert. Die Modellierung der stochastischen Eingangsparameter durch epsilon-korrelierte Felder gestattet Entwicklungen der Lösungscharakteristiken nach der Korrelationslänge. Numerische Beispiele enthalten den Vergleich zwischen analytischen Ergebnissen und Simulationsresultaten.

In the present paper we suggest the norm-preserving explicit operator for the extension of finite-element functions from boundaries of domains into the inside. The construction of this operator is based on the multilevel decomposition of functions on the boundaries and on the equivalent norm for this decomposition. The cost of the action of this operator is proportional to the number of nodes.

In dieser Arbeit werden parabolische Randanfangswertprobleme mit zufälliger Anfangs- und Neumann-Randbedingung betrachtet. Die zufälligen Einflußgrößen werden dabei als epsilon-korrelierte, zufällige Felder modelliert. Das Hauptinteresse liegt auf der Berechnung stochastischer Kenngrößen der auf Basis der Finite-Elemente Methode erhaltenen Lösung des Randanfangswertproblems. Für die Korrelationsfunktion der Lösung wird eine Entwicklung nach der Korrelationslänge sowie eine explizite Berechnung für spezielle Typen der Vernetzung vorgestellt. Anhand von numerischen Beispielen werden abschließend die auf den verschiedenen Wegen erhaltenen Varianzen mit der einer simulierten Lösung verglichen.

This paper deals with the visualization of numerical results as a very convenient method to understand and evaluate a solution which has been calculated as a set of millions of numerical values. One of the central research fields of the Chemnitz SFB 393 is the analysis of parallel numerical algorithms for large systems of linear equations arising from differential equations (e.g. in solid and fluid mechanics). Solving large problems on massively parallel computers makes it more and more impossible to store numerical data from the distributed memory of the parallel computer to the disk for later postprocessing. However, the developer of algorithms is interested in an on-line response of his algorithms. Both visual and numerical response of the running program may be evaluated by the user for a decision how to switch or adjust interactively certain parameters that may influence the solution process. The paper gives a survey of current programmer and user interfaces that are used in our various 2D and 3D parallel finite element programs for the visualization of the solution.

In this paper, finite element discretizations of the degenerated operator -ω²(y) u_xx-ω²(x)u_yy=g in the unit square are investigated, where the weight function satisfies ω(ξ)=ξ^α with α ≥ 0. We propose two multi-level methods in order to solve the resulting system of linear algebraic equations. The first method is a multi-grid algorithm with line-smoother. A proof of the smoothing property is given. The second method is a BPX-like preconditioner which we call MTS-BPX preconditioner. We show that the upper eigenvalue bound of the MTS-BPX preconditioned system matrix grows proportionally to the level number.

Die vorliegende Arbeit befasst sich mit verschiedenen Aspekten der hp-FEM. Insbesondere werden hierbei folgende Punkte genauer untersucht: 1. Das effiziente Aufstellen der Steifigkeitsmatrix auf Referenzelementen, die keine Tensorproduktstruktur besitzen. 2. Eine lokale Konvergenzbetrachtung für die randkonzentrierte hp-FEM. 3. Ein Multilevel-Löser für die randkonzentrierte hp-FEM. 4. Eine Strategie für hp-Adaptivität.

In this paper, a mixed boundary value problem for the Laplace-Beltrami operator is considered for spherical domains in $R^3$, i.e. for domains on the unit sphere. These domains are parametrized by spherical coordinates (\varphi, \theta), such that functions on the unit sphere are considered as functions in these coordinates. Careful investigation leads to the introduction of a proper finite element space corresponding to an isotropic triangulation of the underlying domain on the unit sphere. Error estimates are proven for a Clément-type interpolation operator, where appropriate, weighted norms are used. The estimates are applied to the deduction of a reliable and efficient residual error estimator for the Laplace-Beltrami operator.

Singularly perturbed reaction-diffusion problems exhibit in general solutions with anisotropic features, e.g. strong boundary and/or interior layers. This anisotropy is reflected in the discretization by using meshes with anisotropic elements. The quality of the numerical solution rests on the robustness of the a posteriori error estimator with respect to both the perturbation parameters of the problem and the anisotropy of the mesh. The simplest local error estimator from the implementation point of view is the so-called hierarchical error estimator. The reliability proof is usually based on two prerequisites: the saturation assumption and the strengthened Cauchy-Schwarz inequality. The proofs of these facts are extended in the present work for the case of the singularly perturbed reaction-diffusion equation and of the meshes with anisotropic elements. It is shown that the constants in the corresponding estimates do neither depend on the aspect ratio of the elements, nor on the perturbation parameters. Utilizing the above arguments the concluding reliability proof is provided as well as the efficiency proof of the estimator, both independent of the aspect ratio and perturbation parameters.

The development of adaptive finite element procedures for the solution of geometrically and physically nonlinear problems in structural mechanics is very important for the augmentation of the efficiency of FE-codes. In this contribution methods of mesh refinement as well as mesh coarsening are presented for a material model considering finite elasto-plastic deformations. For newly generated elements stresses, strains and internal variables have to be calculated. This implies the determination of the nodal values as well as the Gaussian point values of the new elements based on the transfer of data from the former mesh. Analogously, the coarsening of less important elements necessitates the determination of these values for the newly created father elements.