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**Finite elements. Theory and algorithms.**
*(English)*
Zbl 1382.65396

Cambridge-IISc Series. Delhi: Cambridge University Press (ISBN 978-1-108-41570-5/hbk; 978-1-108-23501-3/ebook). viii, 208 p. (2017).

This book presents the mathematical theory and algorithms of the finite element method, a widely-used method for the solution of partial differential equations in the field of Computer Science. It is based on courses given by the authors for many years.

Chapter 1 is a brief introduction to Sobolev spaces and functional analysis.

Chapter 2 describes the finite element method for beginners and introduces concepts of weak solutions, variational formulation of second-order elliptic boundary value problems, incorporation of different boundary value problems in a variational form and the standard Galerkin approach.

Chapter 3 discusses the construction of finite elements on simplices, quadrilaterals and hexahedrals. Linear, bilinear and isoparametric transformations are explained and mapped finite elements are considered.

Chapter 4 deals with interpolation of affine equivalent finite elements in Sobolev spaces.

Chapters 5 and 6 describe finite elements for more advanced scalar problems. Chapter 5 studies conforming and nonconforming finite element methods for the biharmonic equation. The finite element method for scalar parabolic problems is presented in Chapter 6.

Finite element methods for systems of equations in solid mechanics and fluid mechanics are the subjects of Chapters 7 and 8.

Finally, in Chapter 9, finite element algorithms and implementation based on object-oriented concepts are provided.

The book is written in such a way that the theoretic material of Chapters 1 to 4 and the implementation matters of Chapter 9 should be presented in any course on finite elements. The selection of other chapters is a matter of taste.

Chapter 1 is a brief introduction to Sobolev spaces and functional analysis.

Chapter 2 describes the finite element method for beginners and introduces concepts of weak solutions, variational formulation of second-order elliptic boundary value problems, incorporation of different boundary value problems in a variational form and the standard Galerkin approach.

Chapter 3 discusses the construction of finite elements on simplices, quadrilaterals and hexahedrals. Linear, bilinear and isoparametric transformations are explained and mapped finite elements are considered.

Chapter 4 deals with interpolation of affine equivalent finite elements in Sobolev spaces.

Chapters 5 and 6 describe finite elements for more advanced scalar problems. Chapter 5 studies conforming and nonconforming finite element methods for the biharmonic equation. The finite element method for scalar parabolic problems is presented in Chapter 6.

Finite element methods for systems of equations in solid mechanics and fluid mechanics are the subjects of Chapters 7 and 8.

Finally, in Chapter 9, finite element algorithms and implementation based on object-oriented concepts are provided.

The book is written in such a way that the theoretic material of Chapters 1 to 4 and the implementation matters of Chapter 9 should be presented in any course on finite elements. The selection of other chapters is a matter of taste.

Reviewer: I. N. Katz (St. Louis)

### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

65N15 | Error bounds for boundary value problems involving PDEs |