Elman, Howard C.; Silvester, David J.; Wathen, Andrew J. Finite elements and fast iterative solvers. With applications in incompressible fluid dynamics. 2nd ed. (English) Zbl 1304.76002 Numerical Mathematics and Scientific Computation. Oxford: Oxford University Press (ISBN 978-0-19-967879-2/hbk; 978-0-19-967880-8/pbk). xiv, 479 p. (2014). For the review of the first edition see [(2005; Zbl 1083.76001)]. The authors present efficient solutions of Poisson, Stokes, convection-diffusion and Navier-Stokes partial differential equations (PDEs) that arise when modelling incompressible fluid flows. The theoretical results (concerning weak formulation, the Galerkin finite element method, theory of errors, the conjugate gradient method, matrix properties, preconditioning methods, splitting operators, multigrid methods) are developed in a constructive style and illustrated by numerical experiments. Error estimation and iterative solution of linear systems of algebraic equations are placed into the context of solving the particular discrete PDEs for model problems. The computational examples and exercises are performed using the incompressible flow iterative software IFISS which can be run using MATLAB or GNU Octave. The latest version of the software package is available in source code on a web site.The material, except for some mathematical details, is self-contained, and the book is targeted to researchers and advanced undergraduate and graduate students in engineering, numerical analysis, applied mathematics and interdisciplinary scientific computing. The integration of the numerical methods from linear algebra (in particular, Krylov subspace methods and multilevel preconditioners) into the context of solving particular discrete PDEs enables the authors to prove some convergence results. Due to this approach, the book is intended to play a bridge-building role between the PDE and linear algebra research communities. In the second edition of the book, two new chapters have been added concerning the unsteady Navier-Stokes equations and the models of buoyancy-driven flows. The chapter devoted to the discrete steady Navier-Stokes equations has also been substantially revised. Reviewer: Adrian Carabineanu (Bucureşti) Cited in 4 ReviewsCited in 316 Documents MSC: 76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics 76M10 Finite element methods applied to problems in fluid mechanics 76D05 Navier-Stokes equations for incompressible viscous fluids 76D07 Stokes and related (Oseen, etc.) flows 76R99 Diffusion and convection 65F10 Iterative numerical methods for linear systems Keywords:Poisson equation; Galerkin finite element method; Stokes equations; convection-diffusion equations; Navier-Stokes equations; buoyancy-driven flow; preconditioning; convergence; error bounds Citations:Zbl 1083.76001 Software:Matlab; IFISS; Octave PDFBibTeX XMLCite \textit{H. C. Elman} et al., Finite elements and fast iterative solvers. With applications in incompressible fluid dynamics. 2nd ed. Oxford: Oxford University Press (2014; Zbl 1304.76002) Full Text: DOI