Sobolev Spaces and Their Applications

The introductory section (Preliminaries) covers certain concepts from mathematical analysis and functional analysis that the reader should be familiar with before proceeding to study the book itself. Part I is entitled Theory of Sobolev Spaces. It is divided into 20 chapters and covers all the knowledge necessary for understanding the mathematical theory of the finite element method and for working within this theory. Part II, although entitled Basic applications of the theory of Sobolev spaces in the finite element method, contains in Chapter 26 the proof of the most difficult theorem presented in the book. Part III, ‘Variational problems in bad domains’, contains some of the author’s results on the finite element method in two-dimensional domains whose boundaries have so-called turning points. Part IV Variational problems in the case of parabolic-elliptic equations contains some of the author’s results which have applications in quasi-stationary problems of electromagnetic fields, particularly in the theory of rotating electrical machines Part V Semiregular finite elements. This chapter discusses certain aspects of so-called semiregular elements, where, in the case of triangular elements, only the maximum angle condition is satisfied, according to which the largest angle of the triangle must not approach ?. This means that the triangles may have one angle of arbitrarily small size. This section also explains the fundamental difficulties that arise in the theory of the finite element method when the minimum angle condition is not satisfied.