Theory and practice of finite elements. (English) Zbl 1059.65103

Ern, Alexandre; Guermond, Jean-Luc

Applied Mathematical Sciences 159. New York, NY: Springer (ISBN 0-387-20574-8/hbk). xiii, 524 p. (2004).

This book is an expanded version of Lecture Notes published by the authors in French (2002; Zbl 0993.65123). It has been used as a textbook for graduate finite element courses at various French universities and at the University of Texas in Austin. The book can be used in several courses in Mathematics, Computer Science, and Engineering programs. The authors offer suggestions for course titles and syllabi.
The book is organized into three parts. The first part (Chapters 1 and 2) reviews the theoretical foundations of the finite element method. The second and third parts are devoted to the practice of finite elements. The second part (Chapters 3 to 6) deals with partial differential equation based applications of finite elements. The third part (Chapters 7 to 10) covers implementation aspects. Two appendices summarize the main mathemaical concepts used in this book: Banach and Hilbert spaces; Banach operators: distributions; and Sobolev spaces.
This book is not an exhaustive monograph and always tries to remain at the graduate level. Some aspects of the finite element method are only briefly mentioned or simply alluded to (e.g., the \(p\)- and \(hp\)-versions of the method and the use of hierachical settings). Many bibliographic entries to the extensive literature on finite elements are given throughout the book.

Reviewer: I. N. Katz (St. Louis)

MSC:

65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N50	Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
35A35	Theoretical approximation in context of PDEs
65F10	Iterative numerical methods for linear systems
65M60	Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
74S05	Finite element methods applied to problems in solid mechanics
76D07	Stokes and related (Oseen, etc.) flows
35J25	Boundary value problems for second-order elliptic equations
35K15	Initial value problems for second-order parabolic equations
35Q30	Navier-Stokes equations
65-02	Research exposition (monographs, survey articles) pertaining to numerical analysis
76M10	Finite element methods applied to problems in fluid mechanics
65Y05	Parallel numerical computation
65M15	Error bounds for initial value and initial-boundary value problems involving PDEs
65N12	Stability and convergence of numerical methods for boundary value problems involving PDEs
65M12	Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
78M10	Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
80M10	Finite element, Galerkin and related methods applied to problems in thermodynamics and heat transfer

Keywords:

finite elements; textbook; interpolation; mixed methods; error bounds; incompressible fluids; Stokes problem; linear elasticity; boundary value problems; Timoshenko beam; advection-diffusion equation; algorithms; Darcy equation; Maxwell equation; parabolic problems; evolution equations; mesh generation; iterative methods; parallel computation; stability; numerical examples

Citations:

Zbl 0993.65123