Finite element methods and their applications. (English) Zbl 1082.65118

Scientific Computation. Berlin: Springer (ISBN 3-540-24078-0/hbk). xiii, 410 p. (2005).
The first six chapters of the book are concerned with general finite element methods or general classes of elements. Chapter 1 contains the general theory on conforming finite elements including the introduction of Sobolev spaces. The solution of the resulting linear systems by Gaussian elimination and the conjugate gradient algorithm are discussed. Chapter 2 “Nonconforming finite elements” presents finite elements with continuity at midpoints of edges or faces and finite element for partial differential equations (PDEs) of fourth order. Chapter 3 “Mixed finite elements” focuses on finite elements for the space \(H(\text{div})\) since the author wants a simple setting of mixed methods. Examples of iterative methods for saddle point problems are added. Chapter 4 on discontinuous Galerkin methods with and without penalty terms gets its motivation from the treatment of advection problems. Chapter 5 “Characteristic finite elements” is devoted to time dependent PDEs. Chapter 6 “Adaptive finite elements” refers to four ways of a posteriori error estimates and discusses the data management of refined grids.
The last four chapters are directed to some applications. Chapter 7 “Solid mechanics” provides the equations of linear elasticity and a brief presentation of finite elements in this context. Chapter 8 “Fluid mechanics” is also very short since there is much literature on Stokes and Navier-Stokes equations. Chapter 9 “Fluid flow in porous media” with an elaboration of two-phase flow and Chapter 10 “Semiconductor modeling” that starts from Boltzmann equations show that the author is interested very much in these two topics.
The book is based on the material that the author has used in a graduate course for several years. The attempt to introduce every concept in the simplest possible setting is reflected in the structure of the chapters. In chapters 2 to 10 a Section “Theoretical considerations” is put at the end of the chapters immediately before the bibliographical remarks and the exercises. The variational problems of solid mechanics are discussed without mentioning Korn’s inequality. [In the formula for the nonlinear strains there is the same error as in the referee’s book on finite elements: Finite Elemente. Theorie, schnelle Löser und Anwendungen in der Elastizitätstheorie. (1992; Zbl 0754.65084)].
The book and its intention differ very much from the books on finite elements by S. C. Brenner and L. R. Scott [The mathematical theory of finite element methods. Texts in Applied Mathematics. 15, New York, Springer (1994; Zbl 0804.65101)] or the referee. The reader finds here more variants of finite element spaces and applications that have not been described in textbooks on finite elements and in particular not with so many details. So the referee wishes the author that his book also finds its readers.


65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
74S05 Finite element methods applied to problems in solid mechanics
74B05 Classical linear elasticity
76D05 Navier-Stokes equations for incompressible viscous fluids
76D07 Stokes and related (Oseen, etc.) flows
76M10 Finite element methods applied to problems in fluid mechanics
65F05 Direct numerical methods for linear systems and matrix inversion
65F10 Iterative numerical methods for linear systems
35Q30 Navier-Stokes equations
76S05 Flows in porous media; filtration; seepage
82-08 Computational methods (statistical mechanics) (MSC2010)
82D37 Statistical mechanics of semiconductors


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