Survey of Variational Principles and Associated Finite Element Methods. -- Classical Variational Methods -- Alternative Variational Formulations -- Abstract Theory of Least-Squares Finite Element Methods -- Mathematical Foundations of Least-Squares Finite Element Methods -- The Agmon–Douglis–Nirenberg Setting for Least-Squares Finite Element Methods -- Least-Squares Finite Element Methods for Elliptic Problems -- Scalar Elliptic Equations -- Vector Elliptic Equations -- The Stokes Equations -- Least-Squares Finite Element Methods for Other Settings -- The Navier–Stokes Equations -- Parabolic Partial Differential Equations -- Hyperbolic Partial Differential Equations -- Control and Optimization Problems -- Variations on Least-Squares Finite Element Methods -- Supplementary Material -- Analysis Tools -- Compatible Finite Element Spaces -- Linear Operator Equations in Hilbert Spaces -- The Agmon–Douglis–Nirenberg Theory and Verifying its Assumptions.

The book examines theoretical and computational aspects of least-squares finite element methods(LSFEMs) for partial differential equations (PDEs) arising in key science and engineering applications. It is intended for mathematicians, scientists, and engineers interested in either or both the theory and practice associated with the numerical solution of PDEs. The first part looks at strengths and weaknesses of classical variational principles, reviews alternative variational formulations, and offers a glimpse at the main concepts that enter into the formulation of LSFEMs. Subsequent parts introduce mathematical frameworks for LSFEMs and their analysis, apply the frameworks to concrete PDEs, and discuss computational properties of resulting LSFEMs. Also included are recent advances such as compatible LSFEMs, negative-norm LSFEMs, and LSFEMs for optimal control and design problems. Numerical examples illustrate key aspects of the theory ranging from the importance of norm-equivalence to connections between compatible LSFEMs and classical-Galerkin and mixed-Galerkin methods. Pavel Bochev is a Distinguished Member of the Technical Staff at Sandia National Laboratories with research interests in compatible discretizations for PDEs, multiphysics problems, and scientific computing. Max Gunzburger is Frances Eppes Professor of Scientific Computing and Mathematics at Florida State University and recipient of the W.T. and Idelia Reid Prize in Mathematics from the Society for Industrial and Applied Mathematics.