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The finite element method. Theory, implementation, and applications. (English) Zbl 1263.65116

Texts in Computational Science and Engineering 10. Berlin: Springer (ISBN 978-3-642-33286-9/hbk; 978-3-642-33287-6/ebook). xvii, 385 p. (2013).
The authors give an introduction to the finite element method as a general computational method for solving partial differential equations (PDEs) approximately. The authors mix the mathematical theory with a concrete computer code using the numerical software Matlab and its PDE toolbox. A basic knowledge of the Matlab language is necessary. The PDE toolbox is used for pre and post processing. The authors also study some applications of finite elements, such as diffusion and transport phenomena, solid and fluid mechanics and electromagnetics.
The book is divided in two parts. Chapters 1-6 provide basic material and chapters 7-15 offer advanced material and applications. In the first part, Chapters 1-4 give an introduction to the finite element method for stationary second-order elliptic problems. In Chapter 5, the authors consider time-dependent problems, and in Chapter 6, they give an introduction to numerical linear algebra for sparse linear systems of equations.
The abstract theory is presented in Chapter 7 and in Chapter 8, the authors present various finite elements. Chapter 9 discusses a short introduction to nonlinear problems. Chapters 10-13 present applications to transport problems, solid mechanics, fluid mechanics and electromagnetics. Chapter 14 gives a short introduction to discontinuous Galerkin methods.
The book should be accessible to students with only knowledge of calculus of several variables, basic partial differential equations, and linear algebra, as the necessary concepts from more advanced analysis are introduced when needed.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
74S05 Finite element methods applied to problems in solid mechanics
76M10 Finite element methods applied to problems in fluid mechanics
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
35J25 Boundary value problems for second-order elliptic equations
35Q35 PDEs in connection with fluid mechanics
35Q60 PDEs in connection with optics and electromagnetic theory
35Q74 PDEs in connection with mechanics of deformable solids
65Y15 Packaged methods for numerical algorithms
65F50 Computational methods for sparse matrices
65F10 Iterative numerical methods for linear systems
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