Galerkin finite element methods for parabolic problems.
2nd revised and expanded ed.

*(English)*Zbl 1105.65102
Berlin: Springer (ISBN 3-540-33121-2/hbk). xii, 370 p. (2006).

The present monograph presents an essentially self-contained account of the mathematical theory of Galerkin finite element methods as applied to parabolic equations. This edition is an updated and extended version of its first edition (1997; Zbl 0884.65097).

The new ideas are mainly concerned with the application of semigroup theory to stability and error analysis. The author has revised the 18 chapters of the textbook and has added 2 new chapters, namely on polygonal domains and on time discretization by Laplace transformation and quadrature.

In Chapter 1, the author recalls standard material from the finite element method (FEM) for elliptic problems and investigates the (spatially) semidiscrete and some fully discrete schemes for some simple model initial-boundary value problem. Chapters 2-6 are devoted to several extensions and generalizations of the results in the semidiscrete case. Chapter 6 contains an expanded review of analytic semigroups on maximum norm estimates for the semidiscrete problem, where now resolvent estimates for piecewise linear finite elements are discussed in detail.

The next six chapters are concerned with fully discrete schemes. Chapter 9 and 10 are also altered on behalf of the semigroup theory.

Chapters 13 and 14 are devoted to nonlinear problems. In the next four chapters the author considers various modifications of the standard Galerkin FEM, namely mass lumping (Chap. 15), the \(H^1\) and \(H^{-1}\) methods (Chap. 16), a mixed method (Chap. 17) and, finally, a singular problem (Chap 18).

In Chapter 19 the author discusses problems in polygonal domains with particular attention given to nonconvex domanins. The last chapter presents an alternative to time stepping as a method for discretization in time. The new method is based on representing the solution as an integral involving the resolvent of the elliptic operator along a smooth curve extending into the right half of the complex plane, and then applying the accurate quadrature rule to this integral.

The new ideas are mainly concerned with the application of semigroup theory to stability and error analysis. The author has revised the 18 chapters of the textbook and has added 2 new chapters, namely on polygonal domains and on time discretization by Laplace transformation and quadrature.

In Chapter 1, the author recalls standard material from the finite element method (FEM) for elliptic problems and investigates the (spatially) semidiscrete and some fully discrete schemes for some simple model initial-boundary value problem. Chapters 2-6 are devoted to several extensions and generalizations of the results in the semidiscrete case. Chapter 6 contains an expanded review of analytic semigroups on maximum norm estimates for the semidiscrete problem, where now resolvent estimates for piecewise linear finite elements are discussed in detail.

The next six chapters are concerned with fully discrete schemes. Chapter 9 and 10 are also altered on behalf of the semigroup theory.

Chapters 13 and 14 are devoted to nonlinear problems. In the next four chapters the author considers various modifications of the standard Galerkin FEM, namely mass lumping (Chap. 15), the \(H^1\) and \(H^{-1}\) methods (Chap. 16), a mixed method (Chap. 17) and, finally, a singular problem (Chap 18).

In Chapter 19 the author discusses problems in polygonal domains with particular attention given to nonconvex domanins. The last chapter presents an alternative to time stepping as a method for discretization in time. The new method is based on representing the solution as an integral involving the resolvent of the elliptic operator along a smooth curve extending into the right half of the complex plane, and then applying the accurate quadrature rule to this integral.

Reviewer: Răzvan Răducanu (Iaşi)

##### MSC:

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

35K15 | Initial value problems for second-order parabolic equations |

35K55 | Nonlinear parabolic equations |

65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |