Dietrich Braess - Böcker

This definitive introduction to finite element methods was thoroughly updated for this 2007 third edition, which features important material for both research and application of the finite element method. The discussion of saddle-point problems is a highlight of the book and has been elaborated to include many more nonstandard applications. The chapter on applications in elasticity now contains a complete discussion of locking phenomena. The numerical solution of elliptic partial differential equations is an important application of finite elements and the author discusses this subject comprehensively. These equations are treated as variational problems for which the Sobolev spaces are the right framework. Graduate students who do not necessarily have any particular background in differential equations, but require an introduction to finite element methods will find this text invaluable. Specifically, the chapter on finite elements in solid mechanics provides a bridge between mathematics and engineering.

Bei der numerischen Behandlung partieller Differentialgleichungen treten oft überraschende Phänomene auf. Neben der zügigen Behandlung der klassischen Theorie, die bis an die aktuelle Forschung heranführt, wird deshalb viel Wert auf die Darstellung von Beispielen und Gegenbeispielen gelegt. Die Beispiele haben mit dazu beigetragen, dass das Buch jetzt zu den Standardwerken bei den Finiten Elementen zählt.Mit der fünften Auflage erfolgte eine weitere Abrundung bei den Themen, deren Bedeutung  in den letzten Jahren gewachsen ist. Mit der Theorie der a posteriori Fehlerschätzer wird  a priori Information über den Diskretisierungsfehler  gewonnen, die in der klassischen Theorie noch nicht hergeleitet wurden und die – schärfer als sonst –eine Eigenart von a posteriori Schätzern beleuchtet. Die Behandlung von Platten in der Festkörpermechanik erhält jetzt mit dem Zwei-Energien-Prinzip eine solide Grundlage, nachdem in der letzten Auflage die Behandlung von Locking Effekten in einevollständige Theorie mündete.Das Buch richtet sich an Studierende der Mathematik im 3. Und 4. Studienjahr und in den späteren Kapiteln auch an junge Forscher, bei denen Finite Elemente im Mittelpunkt ihrer Arbeit stehen.

The first investigations of nonlinear approximation problems were made by P.L. Chebyshev in the last century, and the entire theory of uniform approxima­ tion is strongly connected with his name. By making use of his ideas, the theories of best uniform approximation by rational functions and by polynomials were developed over the years in an almost unified framework. The difference between linear and rational approximation and its implications first became apparent in the 1960's. At roughly the same time other approaches to nonlinear approximation were also developed. The use of new tools, such as nonlinear functional analysis and topological methods, showed that linearization is not sufficient for a complete treatment of nonlinear families. In particular, the application of global analysis and the consideration of flows on the family of approximating functions intro­ duced ideas which were previously unknown in approximation theory. These were and still are important in many branchesof analysis. On the other hand, methods developed for nonlinear approximation prob­ lems can often be successfully applied to problems which belong to or arise from linear approximation. An important example is the solution of moment problems via rational approximation. Best quadrature formulae or the search for best linear spaces often leads to the consideration of spline functions with free nodes. The most famous problem of this kind, namely best interpolation by poly­ nomials, is treated in the appendix of this book.