Sobolev Gradients and Differential Equations

• Main Author: neuberger, john (Author, http://id.loc.gov/vocabulary/relators/aut)
• Corporate Author: SpringerLink (Online service)
• Format: Electronic eBook
• Language: English
• Published: Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1997.
• Edition: 1st ed. 1997.
• Series: Lecture Notes in Mathematics, 1670
• Subjects: Partial differential equations.; Numerical analysis.; Partial Differential Equations.; Numerical Analysis.

A Sobolev gradient of a real-valued functional is a gradient of that functional taken relative to the 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. Equal emphasis is placed on numer...

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 gradients 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.