The book "Spectral Methods: Fundamentals in Single Domains" presents a comprehensive overview of spectral methods, which are high-order numerical techniques used for solving partial differential equations. The authors, Claudio Canuto, M. Yousuff Hussaini, Alfio Quarteroni, and Thomas A. Zang, provide a detailed discussion of the theory, algorithms, and applications of spectral methods. The book is structured into two volumes, with the first focusing on the general theory and the second on applications in fluid dynamics and multidomain spectral methods. Spectral methods have a long history, dating back to their original proposal in 1944 by Blinova and first implementation in 1954 by Silberman. They were later revived in the 1960s and 1970s, with significant theoretical developments in the 1980s and 1990s. The book discusses the evolution of spectral methods, their mathematical foundations, and their application in various fields such as fluid dynamics, solid mechanics, and financial engineering. The first volume covers the fundamentals of spectral methods, including polynomial approximation, orthogonal polynomials, and the construction of spectral methods. It also discusses the theory of stability and convergence, as well as the application of spectral methods to model boundary-value problems. The second volume focuses on the application of spectral methods to fluid dynamics, including the analysis of linear and nonlinear stability, incompressible flows, and compressible flows. It also discusses multidomain spectral methods, which are used for problems in complex geometries. The book provides a detailed discussion of various spectral methods, including Galerkin, collocation, and tau methods. It also addresses the challenges of applying spectral methods to discontinuous problems and the development of new techniques for handling such cases. The authors also discuss the use of preconditioning, which is essential for the efficient solution of large systems of equations arising from spectral methods. The book is accompanied by a companion volume, "Spectral Methods: Fundamentals in Single Domains," which provides a more detailed discussion of the applications of spectral methods in fluid dynamics and multidomain spectral methods. The authors emphasize the importance of spectral methods in modern computational science and their ability to provide high accuracy with relatively low computational cost. The book is a valuable resource for researchers and practitioners in the field of computational science and engineering.The book "Spectral Methods: Fundamentals in Single Domains" presents a comprehensive overview of spectral methods, which are high-order numerical techniques used for solving partial differential equations. The authors, Claudio Canuto, M. Yousuff Hussaini, Alfio Quarteroni, and Thomas A. Zang, provide a detailed discussion of the theory, algorithms, and applications of spectral methods. The book is structured into two volumes, with the first focusing on the general theory and the second on applications in fluid dynamics and multidomain spectral methods. Spectral methods have a long history, dating back to their original proposal in 1944 by Blinova and first implementation in 1954 by Silberman. They were later revived in the 1960s and 1970s, with significant theoretical developments in the 1980s and 1990s. The book discusses the evolution of spectral methods, their mathematical foundations, and their application in various fields such as fluid dynamics, solid mechanics, and financial engineering. The first volume covers the fundamentals of spectral methods, including polynomial approximation, orthogonal polynomials, and the construction of spectral methods. It also discusses the theory of stability and convergence, as well as the application of spectral methods to model boundary-value problems. The second volume focuses on the application of spectral methods to fluid dynamics, including the analysis of linear and nonlinear stability, incompressible flows, and compressible flows. It also discusses multidomain spectral methods, which are used for problems in complex geometries. The book provides a detailed discussion of various spectral methods, including Galerkin, collocation, and tau methods. It also addresses the challenges of applying spectral methods to discontinuous problems and the development of new techniques for handling such cases. The authors also discuss the use of preconditioning, which is essential for the efficient solution of large systems of equations arising from spectral methods. The book is accompanied by a companion volume, "Spectral Methods: Fundamentals in Single Domains," which provides a more detailed discussion of the applications of spectral methods in fluid dynamics and multidomain spectral methods. The authors emphasize the importance of spectral methods in modern computational science and their ability to provide high accuracy with relatively low computational cost. The book is a valuable resource for researchers and practitioners in the field of computational science and engineering.