Mathematical Foundations of Finite Elements and Iterative Solvers

—    Richard S. Falk, Rutgers University   

“Students of applied mathematics, engineering, and science will welcome this insightful and carefully crafted introduction to the mathematics of finite elements and to algorithms for iterative solvers. Concise, descriptive, and entertaining, the text covers all of the key mathematical ideas and concepts dealing with finite element approximations of problems in mechanics and physics governed by partial differential equations while interweaving basic concepts on Sobolev spaces and basic theorems of functional analysis presented in an effective tutorial style.”

—    J. Tinsley Oden, The University of Texas at Austin 

This textbook describes the mathematical principles of the finite element method, a technique that turns a (linear) partial differential equation into a discrete linear system, often amenable to fast linear algebra. Reflecting the author’s decade of experience in the field, Mathematical Foundations of Finite Elements and Iterative Solvers examines the crucial interplay between analysis, discretization, and computations in modern numerical analysis; furthermore, it recounts historical developments leading to current state-of-the-art techniques. While self-contained, this textbook provides a clear and in-depth discussion of several topics, including elliptic problems, continuous Galerkin methods, iterative solvers, advection-diffusion problems, and saddle point problems. 

Accessible to readers with a beginning background in functional analysis and linear algebra, this text can be used in graduate-level courses on advanced numerical analysis, data science, numerical optimization, and approximation theory. Professionals in numerical analysis and finite element methods will also find the book of interest.