Part I. Survey of Variational Principles and Associated Finite Element Methods. Classical Variational Methods. Alternative Variational Formulations.- Part II. Abstract Theory of Least-Squares Finite Element Methods. Mathematical Foundations. First-Order Agmon-Douglis-Nirenberg Systems.- Part III. Least-Squares Methods for Elliptic Problems. Basic First-Order Systems. Application to Key Elliptic Problems.- Part IV. Extensions of Least-Squares Methods to other Problems. The Navier-Stokes Equations. Dissipative Time Dependent Problems. Hyperbolic Problems. Control and optimization Problems. Other Topics.- Part V. Supplementary Material.- A. Analysis Tools. B. Finite Element Spaces. C. Discrete Norms and Operators. D. The Complementing Condition.
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#x2013;Douglis#x2013;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#x2013;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#x2013;Douglis#x2013;Nirenberg Theory and Verifying its Assumptions.
Since their emergence, finite element methods have taken a place as one of the most versatile and powerful methodologies for the approximate numerical solution of Partial Differential Equations. These methods are used in incompressible fluid flow, heat, transfer, and other problems. This book provides researchers and practitioners with a concise guide to the theory and practice of least-square finite element methods, their strengths and weaknesses, established successes, and open problems.
This book is written to provide a common, mathematically sound foundation for least-squares finite element methods. It is intended to give both the researcher and the practitioner a concise guide to the theory and practice of least-square finite element methods, their strengths and weaknesses, established successes, and open problems.