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

Quarteroni, Alfio; Valli, Alberto

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.

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

Keywords:

algorithms; linear; nonlinear; finite element method; spectral method; finite difference; finite volume methods; convection-diffusion problems; hyperbolic equations; direct and iterative solvers; preconditioning; multigrid; domain decomposition; Navier-Stokes equations