In this paper we develop new preconditioned iterative methods for solving monotone nonlinear eigenvalue problems. We investigate the convergence and derive grid-independent error estimates for these methods. Numerical experiments demonstrate the practical effectiveness of the proposed methods for a model problem.

In this paper we consider the well-known BPX-preconditioner in conjunction with adaptive FEM. We present an algorithm which enables us to compute the preconditioner with optimal complexity by a total of only O(DoF) additional memory. Furthermore, we show how to combine the BPX-preconditioner with an overlapping Additive-Schwarz-preconditioner to obtain a preconditioner for finite element spaces with arbitrary polynomial degree distributions. Numerical examples illustrate the efficiency of the algorithms.

Der Einsatz der Finite-Element-Methode bei linearen piezoelektrischen Problemen führt auf eine Systemmatrix der Struktur \[\left( \begin{array}{lr} C & B \\ B^T & -K \end{array} \right)\] mit positiv definiten Blockmatrizen C und K. Zur Lösung indefiniter Gleichungssysteme, die diese symmetrische Blockstruktur besitzen, kann der Bramble--Pasciak--CG eingesetzt werden. Entscheidend für eine schnelle Lösung ist es dabei, gute Vorkonditionierer für den Block C sowie für ein inexaktes Schurkomplement zu finden. Nachfolgend wird das Schurkomplement auf Spektraläquivalenz zur Blockmatrix K untersucht, für welche gute Vorkonditionierer bekannt sind. / Using the Finite-Element-Method with linear piezoelectric problems leads to a linear system of the structure \[\left( \begin{array}{lr} C & B \\ B^T & -K \end{array} \right)\] with symmetric positive definite matrix blocks C and K. The Bramble--Pasciak--CG is a possible solver for indefinite linear systems of equations with this special symmetric block structure. Essential for fast solving are good preconditioners for the block C as well as for an inexact Schur complement. In the following, the Schur complement is examined to spectral equivalence with the matrix K. For K quite good preconditioners are known.

Systems of grid equations that approximate elliptic boundary value problems on locally modified grids are considered. The triangulation, which approximates the boundary with second order of accuracy, is generated from an initial uniform triangulation by shifting nodes near the boundary according to special rules. This "locally modified" grid possesses several significant features: this triangulation has a regular structure, the generation of the triangulation is rather fast, this construction allows to use multilevel preconditioning (BPX-like) methods. The proposed iterative methods for solving elliptic boundary value problems approximately are based on two approaches: The fictitious space method, i.e. the reduction of the original problem to a problem in an auxiliary (fictitious) space, and the multilevel decomposition method, i.e. the construction of preconditioners by decomposing functions on hierarchical grids. The convergence rate of the corresponding iterative process with the preconditioner obtained is independent of the mesh size. The construction of the grid and the preconditioning operator for the three dimensional problem can be done in the same way.

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 we develop new preconditioned iterative methods for solving monotone nonlinear eigenvalue problems. We investigate the convergence and derive grid-independent error estimates for these methods. Numerical experiments demonstrate the practical effectiveness of the proposed methods for a model problem.

Systems of grid equations that approximate elliptic boundary value problems on locally modified grids are considered. The triangulation, which approximates the boundary with second order of accuracy, is generated from an initial uniform triangulation by shifting nodes near the boundary according to special rules. This "locally modified" grid possesses several significant features: this triangulation has a regular structure, the generation of the triangulation is rather fast, this construction allows to use multilevel preconditioning (BPX-like) methods. The proposed iterative methods for solving elliptic boundary value problems approximately are based on two approaches: The fictitious space method, i.e. the reduction of the original problem to a problem in an auxiliary (fictitious) space, and the multilevel decomposition method, i.e. the construction of preconditioners by decomposing functions on hierarchical grids. The convergence rate of the corresponding iterative process with the preconditioner obtained is independent of the mesh size. The construction of the grid and the preconditioning operator for the three dimensional problem can be done in the same way.

Der Einsatz der Finite-Element-Methode bei linearen piezoelektrischen Problemen führt auf eine Systemmatrix der Struktur \[\left( \begin{array}{lr} C & B \\ B^T & -K \end{array} \right)\] mit positiv definiten Blockmatrizen C und K. Zur Lösung indefiniter Gleichungssysteme, die diese symmetrische Blockstruktur besitzen, kann der Bramble--Pasciak--CG eingesetzt werden. Entscheidend für eine schnelle Lösung ist es dabei, gute Vorkonditionierer für den Block C sowie für ein inexaktes Schurkomplement zu finden. Nachfolgend wird das Schurkomplement auf Spektraläquivalenz zur Blockmatrix K untersucht, für welche gute Vorkonditionierer bekannt sind. / Using the Finite-Element-Method with linear piezoelectric problems leads to a linear system of the structure \[\left( \begin{array}{lr} C & B \\ B^T & -K \end{array} \right)\] with symmetric positive definite matrix blocks C and K. The Bramble--Pasciak--CG is a possible solver for indefinite linear systems of equations with this special symmetric block structure. Essential for fast solving are good preconditioners for the block C as well as for an inexact Schur complement. In the following, the Schur complement is examined to spectral equivalence with the matrix K. For K quite good preconditioners are known.

In this paper we consider the well-known BPX-preconditioner in conjunction with adaptive FEM. We present an algorithm which enables us to compute the preconditioner with optimal complexity by a total of only O(DoF) additional memory. Furthermore, we show how to combine the BPX-preconditioner with an overlapping Additive-Schwarz-preconditioner to obtain a preconditioner for finite element spaces with arbitrary polynomial degree distributions. Numerical examples illustrate the efficiency of the algorithms.

Ischämietoleranz bezeichnet das Phänomen, dass ein kurzer ischämischer, metabolischer oder physikalischer Stimulus das Gehirn paradoxerweise "resistent" macht gegenüber einer darauffolgenden, längerdauernden Ischämie. In einer retrospektiven Studie versuchten wir die Hypothese zu untermauern, dass transiente ischämische Attacken (als kurzdauernde ischämische Stimuli) vor einem Infarkt (prodromale TIAs) protektiv sind gegen eine nachfolgende zerebrale Ischämie. Es zeigte sich dabei, dass Patienten mit prodromalen TIAs ein geringeres Defizit und einen günstigeren Verlauf zeigten und im CT seltener Infarktfrühzeichen hatten. Somit könnten transiente ischämische Attacken, vor einem Schlaganfall, analog zu der Situation am Herzen und wie in zahlreichen in vivo Modellen gezeigt, ein klinisches Korrelat zur hypoxischen Präkonditionierung darstellen. Im experimentellen Teil der vorliegenden Arbeit wird gezeigt, dass sich hypoxische Präkonditionierung in vitro in neuronalen Kulturen modellieren lässt. Eine kurzzeitige Sauerstoff-Glucose-Deprivation (OGD) 1-3 Tage vor einer längeren OGD führt zu einem signifikanten Schutz von Neuronen, bis zu 90%. Hypoxietoleranz kann auch durch andere metabolische Stimuli, wie Inhibition von Atmungskettenenzymen durch 3-NPA im gleichen Zeitrahmen simuliert werden. Eine genaue Kenntnis der endogenen Neuroprotektion durch Ischämietoleranz könnte in Zukunft helfen, den Schaden durch ischämische Infarkte und ischämische Enzephalopathien zu minimieren. / Ischemic tolerance is a phenomenon where a brief episode of ischemia renders the brain resistant against a subsequent, longerlasting ischemic event. In a retrospective study we tested the hypothesis that transient ischemic attacks (as brief ischemic stimuli) before cerebral ischemia (prodromal TIA's) may have a protective effect. Here we show that patients with prodromal TIA's have less severe neurologic impairment, a better clinical course and have less early infarct signs. Therefore we siggest that TIA's, before stroke could represent a clinical correlate to hypoxic preconditioning, as shown in the heart. Experimentally we were able to model hypoxic preconditioning in vitro using neuronal cultures. Brief oxygen-glucose deprivation (OGD) 1-3 days before longer lasting OGD protects neurons, up to 90%. Hypoxic tolerance was also simulated by metabolic stimuli like inhibition of the respiratory chain by 3-NPA. Increasing knowledge of this endogenous neuroprotection by ischemic tolerance might help to minimize neuronal damage following ischemic strokes and hypoxic encephalopathy.