65-XX NUMERICAL ANALYSIS
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Institute
Ziel dieser Arbeit ist die effiziente L\"osung hochdimensionaler elliptischer partieller Differentialgleichungen auf d\"unnen Gittern.
Bei Diskretisierung wird der Galerkin-Ansatz und die Finite-Elemente-Methode verwendet.
Eine effiziente Implementierung der Matrix-Vektor-Multiplikation skaliert linear mit der Anzahl der Gitterpunkte.
Dazu wird ein Algorithmus vorgestellt, der durch eine Kombination von Restriktionen und Prolongationen die hierarchischen \"Ubersch\"usse aller Gitter einsammelt und verteilt.
F\"ur d\"unne Gitter muss allerdings auf den Austausch zwischen einigen Gittern verzichtet werden.
Dies hat aufgrund der Semi-Orthogonalit\"at keinen Einfluss auf die Konvergenz, da Prewavelets f\"ur \"uberlappende Gebiete $L^2$-orthogonal sind.
Bei variablen Koeffizienten zeigt ein Konvergenzbeweis, dass diese Werte sehr klein sind und keinen Einfluss auf die Konvergenz haben.
Die Konvergenzordnung der D\"unngitterdiskretisierung reduziert sich im Vergleich zu vollen Gittern nur leicht, w\"ahrend die Anzahl der Unbekannten dramatisch abnimmt.
Numerische Ergebnisse mit variablen Koeffizienten zeigen optimale Konvergenz f\"ur dreidimensionale und sechsdimensionale Probleme.
Transformationen f\"ur krummlinig umrandete Gebiete besitzen variable Koeffizienten und beruhen nicht auf einem Tensorprodukt.
Die Berechnung der Steifigkeitsmatrix verwendet eine hochdimensionale numerische Integration.
Multiplikationen werden durch Rekursion auf eindimensionale Operatoren zur\"uckgef\"uhrt.
Dazu ist im Rahmen dieser Arbeit eine umfangreiche Softwarebibliothek entstanden.
Durch die Verwendung von Templates kann die Implementierung auf beliebige hochdimensionale Probleme angewendet werden.
Parallelisierung sowohl f\"ur geteilten Speicher als auch verteilte Systeme gew\"ahrleistet eine hohe Genauigkeit f\"ur gro\ss{}e Dimensionen.
Ein Ansatz mit semi-adaptiver Verfeinerung erm\"oglicht partielle Verfeinerung bei gleichzeitigem Erhalt der Tensorprodukt-Struktur.
Die Umsetzung adaptiver d\"unner Gitter wird als Ausblick behandelt.
Nonlocal balance laws are nonlinear partial integro-differential equations that play a major role in the modeling of real world phenomena.
From the description of ripening processes in nanoparticle synthesis up to macroscopic modeling of vehicular traffic flow these equations are of crucial importance.
In the presented work a general formulation of one-dimensional nonlocal balance laws was analyzed with respect to existence, uniqueness and regularity of weak solutions.
The Kruzkov entropy condition is in the context of balance laws widely used to obtain uniqueness of weak solutions. The presented work shows that the entropy condition is obsolete in the discussed class of nonlocal balance laws.
An analytical representation of the weak solution is derived based on the method of characteristics and a fix-point mapping in the function space. The solvability of the fix point mapping for small times is proven by Banach's fixed-point theorem. A time-horizon where the weak solution exists is determined by clustering of these small times steps. The gained result is shown to be sharp in special cases.
Commonly weak solutions to nonlocal balance laws were approximated numerically using problem specific finite volume schemes. To get rid of the inherent numerical dissipation
an alternative numerical scheme is deduced based on the analytical representation of the weak solution. Therefore, a semi-discretization based on the method of characteristics and a piecewise constant representation of the solution is introduced and analyzed with respect to convergence.
In dependence on the global as well as piecewise regularity of the data a priori error estimates are derived. Therefore the whole spectrum from inital data of bounded variation up to Lipschitz-continuous initial data were analyzed.
The presented numerical scheme and parts of the analytical weak solution are applied to examples from the modeling of traffic flow and the nanoparticle synthesis. Thereby the high accuracy of the presented scheme is discussed in comparison to published simulation results. Through the introduced numerical method, discontinuities of the initial data are tracked over time and will not be smoothed, thus the character of the solution is represented more accurately. In the context of nanoparticle synthesis, optimal process conditions which lead to particle size distributions with small dispersity were determined exemplarily.
Liquid-gas-solid three-phase flows play an important role in nature and industrial applications. In this thesis, a direct numerical simulation technique for liquid-gas-solid flows is presented. The model is based on the lattice Boltzmann method for hydrodynamics, a free surface volume-of-fluid approach for liquid-gas interface tracking, and a Lagrangian rigid body representation of solid particles.
The theory of the free surface lattice Boltzmann method is developed in detail, including an analysis of lattice Boltzmann boundary conditions for free surfaces, a comparison of different volume-of-fluid surface tension models, and an analysis of the overall method accuracy. Finally, a coupling to the Lagrangian rigid body simulation is presented to extend the method to liquid-gas-solid flows.
The method is applied to simulate bubble-particle interaction in a containing liquid. Thus, simulations of gas bubbles within particle suspensions become possible. As a test case, a bubble-induced mixing process of up to 178486 fully resolved particles is simulated. Thanks to the scalability of the method on parallel computers, arbitrarily complex setups are possible, if sufficient resources are provided.
The thesis is completed with a self-contained introduction to the lattice Boltzmann method for hydrodynamics. The consistency of the method with the incompressible Navier-Stokes equation is studied by means of Chapman-Enskog analysis.
In this thesis, a-posteriori error estimates for time discretizations of the incompressible time dependent Stokes equations by pressure-correction methods are presented.
Pressure-correction methods are splitting schemes, which decouple the velocity and the
pressure. As a result the Stokes equations reduce to much easier schemes, which can
be solved by cost-efficient algorithm. In a first step, a-posteriori estimates for the instationary Stokes system, discretized by the two-step backward differential formula method
(BDF2), are presented. This allows to compare the a-posteriori estimators of the discretized Stokes system with the estimators of the pressure-correction scheme.
In the second part of the thesis rigorous proofs of global upper bounds for the incremen-
tal pressure correction scheme discretized by backward Euler scheme as well as for the
two-step backward differential formula method (BDF2) in rotational form are presented.
Moreover, rate optimality of the estimators are shown for velocity (in case of backward
Euler and BDF2 in rotational form) and pressure (in case of Euler). Computational
experiments confirm the theoretical results.