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**Numerical approximation of partial differential equations.**
*(English)*
Zbl 0803.65088

Springer Series in Computational Mathematics. 23. Berlin: Springer-Verlag. xvi, 543 p. DM 128.00; öS 998.40; sFr 128.00/hbk (1994).

This is a book on the numerical approximation of partial differential equations. A theoretical analysis, description of algorithms and a discussion of applications are given. Many kinds of problems are addressed: linear and nonlinear, steady and time-dependent, having smooth or non-smooth solutions.

Part I is devoted to different discretizations of partial differential equations. In particular, the finite element method (conforming, non- conforming, mixed, hybrid) and the spectral method (Legendre and Chebyshev expansion) are investigated.

For unsteady problems the finite difference and fractional-step schemes for marching in time are studied. Finite differences and finite volume methods are extensively considered in Parts II and III in the framework of convection-diffusion problems and hyperbolic equations. For the solution of the resulting algebraic systems direct and iterative solvers (with preconditioning) are presented. A short account is also given to multigrid and domain decomposition methods.

The authors consider all classical equations of mathematical physics: elliptic, parabolic and hyperbolic equations. Furthermore, the advection- diffusion and Navier-Stokes equations for viscous incompressible flows are investigated. The general equations of fluid dynamics are derived.

Part I is devoted to different discretizations of partial differential equations. In particular, the finite element method (conforming, non- conforming, mixed, hybrid) and the spectral method (Legendre and Chebyshev expansion) are investigated.

For unsteady problems the finite difference and fractional-step schemes for marching in time are studied. Finite differences and finite volume methods are extensively considered in Parts II and III in the framework of convection-diffusion problems and hyperbolic equations. For the solution of the resulting algebraic systems direct and iterative solvers (with preconditioning) are presented. A short account is also given to multigrid and domain decomposition methods.

The authors consider all classical equations of mathematical physics: elliptic, parabolic and hyperbolic equations. Furthermore, the advection- diffusion and Navier-Stokes equations for viscous incompressible flows are investigated. The general equations of fluid dynamics are derived.

Reviewer: W.Heinrichs (Düsseldorf)

### MSC:

65Mxx | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |

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

65Fxx | Numerical linear algebra |

65H10 | Numerical computation of solutions to systems of equations |

65Nxx | Numerical methods for partial differential equations, boundary value problems |

35Q30 | Navier-Stokes equations |

35Lxx | Hyperbolic equations and hyperbolic systems |

35Kxx | Parabolic equations and parabolic systems |

35Jxx | Elliptic equations and elliptic systems |