Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics

The efficient numerical treatment of incompressible fluid dynamics is a formidable challenge in computational mathematics. To develop an efficient treatment, one needs to understand the basic equations of fluid dynamics; i.e., the Stokes and Navier–Stokes equations, together with the boundary conditions that give wellposedness. Next, for variational formulations, one needs a knowledge of finite element discretizations that are stable or finite element stabilization techniques, in order to generate accurate numerical solutions. Finally, one needs a grasp of fast iterative solution schemes, both linear and nonlinear, so that an overall efficient method can be constructed. To gain acquaintance with such a cutting-edge development, prior to the publication of this book, the main reference sources were technical journal articles on each of these treatment components. Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics, by Elman, Silvester, and Wathen, now gives a thorough presentation of the state-ofthe-art techniques in one volume. The authors are acclaimed in this research field. Indeed, this book presents many of their own research results, but in a readable form accessible to a broad audience of both professionals and students of engineering, mathematics, and interdisciplinary computational science. The book is self-contained. Only basic knowledge of discretization methods for partial differential equations, fundamental functional analysis, and computational linear algebra are needed. Presentation of the basic partial differential equations underlying fluid dynamics, of finite element discretization, and of iterative linear system schemes start from the basics. The book then systematically progresses to more advanced theoretical and computational techniques for incompressible fluid dynamics. These advanced topics include continuous and discrete inf-sup stability and finite elements that satisfy them, stabilized finite element techniques and the patch-test to verify stability, Krylov iterative methods and the theory behind them, construction and justification of efficient preconditioners for the Stokes and Navier–Stokes equations, and a posteriori adaptive procedures. Even though these topics are often accessible only to experts in the field, the authors give a comprehendible yet thorough presentation, with concrete test-suite examples and appropriate analytical and computational exercises to give the reader a grasp of these concepts.