Sobolev Gradients and Differential Equations

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...

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Bibliographic Details
Call Number:Libro Electrónico
Main Author: neuberger, john (Author)
Corporate Author: SpringerLink (Online service)
Format: Electronic eBook
Language:Inglés
Published: Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2010.
Edition:2nd ed. 2010.
Series:Lecture Notes in Mathematics, 1670
Subjects:
Mathematical analysis.
Differential equations.
Numerical analysis.
Analysis.
Differential Equations.
Numerical Analysis.
Online Access:Texto Completo
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
  • Ordinary Differential Equations and Sobolev Gradients
  • Convexity and Gradient Inequalities
  • Boundary and Supplementary Conditions
  • Continuous Newton#x2019;s Method
  • More About Finite Differences
  • Sobolev Gradients for Variational Problems
  • An Introduction to Sobolev Gradients in Non-Inner Product Spaces
  • Singularities and a Simple Ginzburg-Landau Functional
  • The Superconductivity Equations of Ginzburg-Landau
  • Tricomi Equation: A Case Study
  • Minimal Surfaces
  • Flow Problems and Non-Inner Product Sobolev Spaces
  • An Alternate Approach to Time-dependent PDEs
  • Foliations and Supplementary Conditions I
  • Foliations and Supplementary Conditions II
  • Some Related Iterative Methods for Differential Equations
  • An Analytic Iteration Method
  • Steepest Descent for Conservation Equations
  • Code for an Ordinary Differential Equation
  • Geometric Curve Modeling with Sobolev Gradients
  • Numerical Differentiation, Sobolev Gradients
  • Steepest Descent and Newton#x2019;s Method and Elliptic PDE
  • Ginzburg-Landau Separation Problems
  • Numerical Preconditioning Methods for Elliptic PDEs
  • More Results on Sobolev Gradient Problems
  • Notes and Suggestions for Future Work.

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