Sobolev Gradients and Differential Equations

• Main Author: Neuberger, J.W. (Author)
• Corporate Author: SpringerLink (Online service)
• Format: Electronic
• Language: English
• Published: Berlin, Heidelberg : Springer Berlin Heidelberg, 2010.
• Series: Lecture Notes in Mathematics, 1670
• Subjects: Mathematics.; Differential equations, partial.; Numerical analysis.; Partial Differential Equations.; Numerical Analysis.
• Online Access: https://ezaccess.library.uitm.edu.my/login?url=http://dx.doi.org/10.1007/978-3-642-04041-2

A Sobolev gradient of a real-valued functional on a Hilbert space is a gradient of that functional taken relative to an underlying Sobolev norm. This book shows how descent methods using such gradients allow a unified treatment of a wide variety of problems in differential equations. For discrete ve...

Table of Contents:

• Several Gradients
• Comparison of Two Gradients
• Continuous Steepest Descent in Hilbert Space: Linear Case
• Continuous Steepest Descent in Hilbert Space: Nonlinear Case
• Orthogonal Projections, Adjoints and Laplacians
• Introducing Boundary Conditions
• Newton's Method in the Context of Sobolev Gradients
• Finite Difference Setting: the Inner Product Case
• Sobolev Gradients for Weak Solutions: Function Space Case
• Sobolev Gradient in Non-inner Product Spaces: Introduction
• The Superconductivity Equations of Ginzburg-Landau
• Minimal Surfaces
• Flow Problems and Non-inner Product Sobolev Spaces
• Foliations as a Guide to Boundary Conditions
• Some Related Iterative Methods for Differential Equations
• A Related Analytic Iteration Method
• Steepest Descent for Conservation Equations
• A Sample Computer Code with Notes
• Bibliography
• Index.