The subject of this book is the efficient solution of partial differential equations PDEs that arise when modelling incompressible fluid flow.The material is organized into four groups of two chapters each, covering the Poisson equation chapters 1 and 2; the convection-diffusion equation chapters 3 and 4; the Stokes equations chapters 5 and 6; and the Navier-Stokes equations chapters 7 and 8. These equations represent important models within the domain of computational fluid dynamics, but they also arise in many other settings. For each PDE model, there is a chapter concerned with finite element discretization, and a companion chapter concerned with efficient iterative solution of the algebraic equations obtained from discretization. For each problem and associated solvers there is a description of how to compute along with theoretical analysis which guides the choice of approaches. Illustrative numerical results occur throughout the book, which have been computed with the freely downloadable IFISS software. All numerical results should be reproducible by readers who have access to MATLAB and there is considerable scope fopr experimentation in the 'computational laboratory' provided by the software. This book provides an excellent introduction to finite elements, iterative linear solvers and scientific computing aimed at graduates in engineering, numerical analysis, applied mathematics and interdisciplinary scientific computing. Including theoretical problems and practical exercises closely tied with freely downloadable MATLAB software, it is an ideal teaching and learning resource.