This book provides the reader with a suitably detailed, extensive and mathematically precise treatment of spectral/hp element methods applied to numerical solutions of problems in computational fluid dynamics (CFD). It contains a detailed explanation of fundamental concepts, derivation of methods, and their analysis and realization, accompanied by a number of applications. The book comprises ten chapters and six appendices.

The first chapter is concerned with models of compressible and incompressible flow. In chapter 2, fundamental concepts of spectral/hp techniques are formulated in the framework of one-dimensional models. Chapter 3 deals with an extension of fundamental concepts to multidimensional situations. In chapter 4, implementation aspects are treated. In particular, the authors are concerned with the so-called local operations, global operations and pre- and post-processing. In chapter 5, attention is paid to the use of spectral/hp methods for the solution of diffusion equations. It contains discretization of the Helmholtz equation, temporal discretization, investigation of eigenspectra and the iterative solution of the weak Laplacian, and problems in non-smooth domains. Advection and advection-diffusion problems are treated in chapter 6. Here, among other, Galerkin and discontinuous Galerkin techniques are explained.

Chapter 7 is concerned with non-conforming elements. It contains the treatment of interface conditions, constrained approximation, mortar patching and various types of discontinuous Galerkin methods. Chapter 8 is devoted to explanation of algorithms for solution of incompressible flow models based on the use of primitive variables as well as velocity-vorticity formulations. Chapter 9 presents several examples of incompressible flow simulations using simple laminar flow benchmarks, which possess exact solutions and allow one to ensure that spectral/hp codes give the correct results. One also finds here direct numerical simulation (DNS) and large-eddy simulation (LES) of viscous incompressible flow. Finally, in chapter 10, spectral/hp techniques for hyperbolic problems are treated.

The book contains a large amount of material, including a number of exercises, examples and figures. The book will be helpful to specialists coming into contact with CFD, applied and numerical mathematicians, engineers, physicists and specialists in climate and ocean modeling. It can also be recommended for advanced students of these disciplines.

Reviewer:

mf