This book investigates the theoretical underpinnings and practical applications of multigrid methods when applied to finite element formulations. The author, V. V. Shaidurov, provides rigorous justifications for the convergence rates of various multigrid algorithms across a spectrum of problem types, including self-adjoint, non-self-adjoint, positive definite, indefinite, singular, and spectral problems. The text bridges theory with concrete applications, detailing economical triangulation strategies and effective handling of curvilinear boundaries and quasilinear equations. Significant focus is placed on mixed formulations of finite element methods, which are shown to simplify the approximation of complex problems like the biharmonic equation and steady-state Stokes and Navier-Stokes equations.

The book is a specialized text that combines two advanced areas of computational mathematics: finite element methods and multigrid algorithms. Its theoretical rigor in justifying convergence rates and its practical considerations for implementation suggest it is aimed at researchers and advanced students in numerical analysis and scientific computing. The detailed treatment of mixed formulations and specific complex problems indicates a focus on providing robust solutions for challenging engineering and physics applications.