Advances in Numerical Mathematics - Böcker

While wavelets have since their discovery mainly been applied to problems in signal analysis and image compression, their analytic power has more and more also been recognized for problems in Numerical Analysis. Together with the functional analytic framework for different differential and integral quations, one has been able to conceptu­ ally discuss questions which are relevant for the fast numerical solution of such problems: preconditioning issues, derivation of stable discretizations, compression of fully popu­ lated matrices, evaluation of non-integer or negative norms, and adaptive refinements based on A-posteriori error estimators. This research monograph focusses on applying wavelet methods to elliptic differential equations. Particular emphasis is placed on the treatment of the boundary and the boundary conditions. Moreover, a control problem with an elliptic boundary problem as contraint serves as an example to show the conceptual strengths of wavelet techniques for some of the above mentioned issues. At this point, I would like to express my gratitude to several people before and during the process of writing this monograph. Most of all, I wish to thank Prof. Dr. Wolfgang Dahmen to whom I personally owe very much and with whom I have co-authored a large part of my work. He is responsible for the very stimulating and challenging scientific atmosphere at the Institut fiir Geometrie und Praktische Mathematik, RWTH Aachen. We also had an enjoyable collaboration with Prof. Dr. Reinhold Schneider from the Technical University of Chemnitz.

In this work, we study numerical issues related to a common mathematical model which describes ferromagnetic materials, both in a stationary and non­ stationary context. Electromagnetic effects are accounted for in an extended model to study nonstationary magneto-electronics. The last part deals with the numerical analysis of the commonly used Ericksen-Leslie model to study the fluid flow of nematic liquid crystals which find applications in display technologies, for example. All these mathematical models to describe different microstructural phe­ nomena share common features like (i) strong nonlinearities, and (ii) non­ convex side constraints (i.e., I m I = 1, almost everywhere in w C JRd, for the order parameter m : w -+ JRd). One key issue in numerical modeling of such problems is to make sure that the non-convex constraint is fulfilled for computed solutions. We present and analyze different solution strategies to deal with the variational problem of stationary micromagnetism, which builds part I of the book: direct minimization, convexification, and relaxation using Young measure-valued solutions. In particular, we address the following points: • Direct minimization: A spatial triangulation 'generates an artificial exchange energy contribution' in the discretized minimizing problem which may pollute physically relevant exchange energy contributions; its minimizers exhibit multiple scales (with branching structures near the boundary of the ferromagnet) and are difficult to be computed efficiently. We exploit this observation to construct an adaptive scheme which better resolves multiple scale structures (cubic ferromagnets).

Fur die naherungsweise Losung von Randwertproblemen zweiter Ordnung wird eine einheitliche Theorie der Finiten Elemente Methode und der Randelementmethode prasentiert. Neben der Stabilitats- und Fehleranalysis wird vor allem auf effiziente Losungsverfahren eingegangen. Fur die Diskretisierung der auftretenden Randintegraloperatoren werden schnelle Randelementmethoden (Wavelets, Multipol, algebraische Techniken) mit der Darstellung durch partielle Integration verknupft. Durch die Kopplung von FEM und BEM mittels Gebietszerlegungsmethoden konnen gekoppelte Randwertprobleme in komplexen Strukturen behandelt werden. Numerische Beispiele illustrieren die theoretischen Aussagen.

Numerical simulation promises new insight in science and engineering. In ad­ dition to the traditional ways to perform research in science, that is laboratory experiments and theoretical work, a third way is being established: numerical simulation. It is based on both mathematical models and experiments con­ ducted on a computer. The discipline of scientific computing combines all aspects of numerical simulation. The typical approach in scientific computing includes modelling, numerics and simulation, see Figure l. Quite a lot of phenomena in science and engineering can be modelled by partial differential equations (PDEs). In order to produce accurate results, complex models and high resolution simulations are needed. While it is easy to increase the precision of a simulation, the computational cost of doing so is often prohibitive. Highly efficient simulation methods are needed to overcome this problem. This includes three building blocks for computational efficiency, discretisation, solver and computer. Adaptive mesh refinement, high order and sparse grid methods lead to discretisations of partial differential equations with a low number of degrees of freedom. Multilevel iterative solvers decrease the amount of work per degree of freedom for the solution of discretised equation systems. Massively parallel computers increase the computational power available for a single simulation.

Projection methods had been introduced in the late sixties by A. Chorin and R. Teman to decouple the computation of velocity and pressure within the time-stepping for solving the nonstationary Navier-Stokes equations. Despite the good performance of projection methods in practical computations, their success remained somewhat mysterious as the operator splitting implicitly introduces a nonphysical boundary condition for the pressure. The objectives of this monograph are twofold. First, a rigorous error analysis is presented for existing projection methods by means of relating them to so-called quasi-compressibility methods (e.g. penalty method, pressure stabilzation method, etc.). This approach highlights the intrinsic error mechanisms of these schemes and explains the reasons for their limitations. Then, in the second part, more sophisticated new schemes are constructed and analyzed which are exempted from most of the deficiencies of the classical projection and quasi-compressibility methods. '... this book should be mandatory reading for applied mathematicians specializing in computational fluid dynamics.' J.-L.Guermond. Mathematical Reviews, Ann Arbor

Zum Kontext dieses Buches Die numerische Behandlung partieller Differentialgleichungen beinhaltet im allgemeinen die Lösung großer bis sehr großer Gleichungssysteme. Bei dreidimensionalen Problemen z. B. sind mehrere Millionen Unbekannte keine Seltenheit, und obwohl die Rechenleistung der stärksten Computer in den letzten Jahrzehnten exponentiell angestiegen ist, könnten viele praxis­ relevante Probleme heute nicht gelöst werden, wären die Numeriker nicht bei der Entwicklung effizienter Algorithmen ähnlich erfolgreich gewesen. Zu den bemerkenswertesten Fortschritten auf diesem Gebiet zählt die Entwicklung adaptiver Mehrgitter-und Multilevelverfahren, deren Erfolg auf der Verschmelzung zweier leistungsfähiger Konzepte beruht: der Kombination adaptiver Diskretisierungstechniken mit schnellen Mehrgitter- bzw. Multilevellösern. Die Anwendung adaptiver Diskretisierungstechniken dient zunächst dazu, die Anzahl der Unbekannten und damit die Dimension des zu lösenden Gleichungssystems möglichst gering zu halten. Wurden früher zur Diskretisierung partieller Differentialgleichungen in erster Linie gleichmäßig strukturierte Rechteckgitter verwendet, so ist man heute durch den Einsatz ge­ eigneter Fehlerschätzer in der Lage, die Diskretisierung - ausgehend von einem relativ groben Anfangsgitter und einer entsprechend groben Näherungslösung - schrittweise an die aktuel­ le Näherungslösung anzupassen, bis die gewünschte Genauigkeit erreicht ist. Üblicherweise wird dazu das aktuelle Diskretisierungsgitter lokal verfeinert, und zwar an solchen Stellen, wo aufgrund entsprechender Fehlerabschätzungen eine höhere Genauigkeit zu erwarten ist, z. B. in der Nähe von Singularitäten, Grenzschichten, einspringenden Ecken, etc. Bereiche, in denen die Lösung sichals hinreichend glatt herausstellt, bleiben unverfeinert oder könne- etwa bei zeit abhängigen Anwendungen - sogar wieder vergröbert werden.

Andreas Potschka discusses a direct multiple shooting method for dynamic optimization problems constrained by nonlinear, possibly time-periodic, parabolic partial differential equations. In contrast to indirect methods, this approach automatically computes adjoint derivatives without requiring the user to formulate adjoint equations, which can be time-consuming and error-prone. The author describes and analyzes in detail a globalized inexact Sequential Quadratic Programming method that exploits the mathematical structures of this approach and problem class for fast numerical performance. The book features applications, including results for a real-world chemical engineering separation problem.

This work describes a general approach to a posteriori error estimation and adaptive mesh design for ?nite element models where the solution is subjected to inequality constraints. This is an extension to variational inequalities of the so-called Dual-Weighted-Residual method (DWR method) which is based on a variational formulation of the problem and uses global duality arguments for deriving weighted a posteriori error estimates with respect to arbitrary functionals of the error. In these estimates local residuals of the computed solution are multiplied by sensitivity factors which are obtained from a - merically computed dual solution. The resulting local error indicators are used in a feed-back process for generating economical meshes which are tailored - cording to the particular goal of the computation. This method is developed here for several model problems. Based on these examples, a general concept is proposed, which provides a systematic way of adaptive error control for problems stated in form of variational inequalities. F¨ ur Alexandra, Katharina und Merle Contents 1 Introduction 1 2 Models in elasto-plasticity 13 2. 1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . 14 2. 2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3 The dual-weighted-residual method 23 3. 1 A model situation in plasticity . . . . . . . . . . . . . . . . . . 24 3. 2 A posteriori error estimate . . . . . . . . . . . . . . . . . . . . . 25 3. 3 Evaluation of a posteriori error bounds . . . . . . . . . . . . . . 26 3. 4 Strategies for mesh adaptation . . . . . . . . . . . . . . . . . . 28 3. 5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4 Extensions to stabilised schemes 33 4. 1 Discretisation for themembrane-problem . . . . . . . . . . . . 35 4. 2 A posteriori error analysis . . . . . . . . . . . . . . . . . . . . . 37 4. 3 Numerical tests . . . . . . . . . . . . . . . . . . . . . . . . . . .