Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics

This book describes why and how to do Scientific Computing for fundamental models of fluid flow. It contains introduction, motivation, analysis, and algorithms and is closely tied to freely available MATLAB codes that implement the methods described.

The focus is on finite element approximation methods and fast iterative solution methods for the consequent linear(ized) systems arising in important problems that model incompressible fluid flow. The problems addressed are the Poisson equation, Convection-Diffusion problem, Stokes problem and Navier-Stokes problem, including new material on time-dependent problems and models of multi-physics. The corresponding iterative algebra based on preconditioned Krylov subspace and multigrid techniques is for symmetric and positive definite, nonsymmetric positive definite, symmetric indefinite and nonsymmetric indefinite matrix systems respectively. For each problem and associated solvers there is a description of how to compute together with theoretical analysis that guides the choice of approaches and describes what happens in practice in the many illustrative numerical results throughout the book (computed with the freely downloadable IFISS software). All of the numerical results should be reproducible by readers who have access to MATLAB and there is considerable scope for experimentation in the "computational laboratory" provided by the software.

Developments in the field since the first edition was published have been represented in three new chapters covering optimization with PDE constraints (Chapter 5); solution of unsteady Navier-Stokes equations (Chapter 10); solution of models of buoyancy-driven flow (Chapter 11).

Each chapter has many theoretical problems and practical computer exercises that involve the use of the IFISS software. This book is suitable as an introduction to iterative linear solvers or more generally as a model of Scientific Computing at an advanced undergraduate or beginning graduate level.

Howard Elman, Professor of Computer Science, University of Maryland at College Park,David Silvester, Professor, School of Mathematics, University of Manchester,Andy Wathen, Reader in Numerical Analysis, University of Oxford

Howard Elman is a Professor in the Computer Science Department and the Institute for Advanced Computer Studies at the University of Maryland, College Park. He received his doctorate in Computer Science from Yale University in 1982. He has held visiting positions at Stanford University, the University of Manchester Institute of Science and Technology and the University of Oxford. He has served on the editorial boards of SIAM Journal on Scientific Computing, where he was editor-in-chief from 1998-2004, Mathematics of Computation, and Numerical Linear Algebra with Applications. His research concerns numerical solution of partial differential equations, computational fluid dynamics, sparse matrix methods, and uncertainty quantification.


David Silvester is a Professor in the School of Mathematics at The University of Manchester. His research concerns numerical solution of partial differential equations, computational fluid dynamics, uncertainty quantification, and high performance computing. He received his doctorate in Mathematics from the University of Manchester Institute of Science and Technology in 1984 and has had visiting positions at Stanford University, the University of Maryland at College Park, and the Universite du Littoral, France. He has served on the editorial boards of SIAM Journal on Scientific Computing and the International Journal for Numerical Methods in Fluids.


Andy Wathen is Reader in Numerical Analysis at the Oxford University Mathematical Institute, UK and a Fellow at New College. His research focuses on Scientific Computing methods and algorithms associated with the numerical solution of partial differential equations, particularly algorithms of numerical linear algebra. Applications range from coolant flows to biological patterning. He received his doctorate from Reading University in 1982 and has held visiting positions at Stanford University, the University of California, Berkeley and at the University of New South Wales. He has served on the editorial boards of the IMA Journal on Numerical Analysis, SIAM Journal on Scientific Computing, SIAM Journal on Matrix Analysis and Applications, Numerical Linear Algebra with Applications and Electronic Transactions on Numerical Analysis.