Marti J.T. Introduction to Sobolev Spaces and Finite Element Solution of Elliptic Boundary Value Problems

London: Academic Press, 1987. — 224 p.The finite element method is now widely used in many areas of applied mathematics, physics and engineering. The method enables numerical solutions to be calculated for such problems as those involving electrostatic and electromagnetic fields, ideal flow in hydro- and aerodynamics, diffusion processes, temperature distributions and strains and displacements in elastic media. The method is in fact an ingenious application of the Galerkin method to partial differential equations and associated boundary conditions. This enables these to be reduced to algebraic form, which can then be solved as an algebraic equation system. As the great advantage of the finite element method is its use for the practical solution of two- or three-dimensional boundary value problems in irregular domains, it is not surprising that the implementation of the method must invoke and depend on techniques from the fields of computer science and numerical linear algebra. However, an understanding of boundary value problems for partial differential equations and the convergence proofs for finite element methods requires a great deal of insight into the properties of the underlying Sobolev spaces of partial differentiate functions. This vast range of information hinders many an interested engineer or scientist from understanding the method as a whole. This reflects the fact that most monographs in the field of finite elements cover only isolated areas of this fascinatingly broad spectrum of knowledge, which encompasses such areas as Sobolev spaces, the theory of elliptic boundary value problems, the analysis of the finite element method, engineering aspects, data processing and the numerical solution of large sparse systems of linear and nonlinear equations.The object of this book is to give a thorough introduction to a number of theoretical fields connected with the finite element method, which have been mentioned above. It has developed as a result of various lectures on the analysis of the finite element method taught by the author at the ETH in Zurich. Encouraged by the response to these lectures, the author feels that the book has become a readable text which can help ordinary mortals to start to understand the theory of finite element methods. The reader is assumed to have only an undergraduate background in mathematical analysis.