neuberger Sobolev Gradients and Differential Equations

1997
ISBN: 978-3-540-69594-3
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 1670, 152 Seiten, eBook

Reihe: Lecture Notes in Mathematics

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 numerical and theoretical matters. Several concrete applications are made to illustrate the method. These applications include (1) Ginzburg-Landau functionals of superconductivity, (2) problems of transonic flow in which type depends locally on nonlinearities, and (3) minimal surface problems. Sobolev gradient constructions rely on a study of orthogonal projections onto graphs of closed densely defined linear transformations from one Hilbert space to another. These developments use work of Weyl, von Neumann and Beurling.

Zielgruppe

Research

Autoren/Hrsg.

neuberger, john

Inhaltsverzeichnis

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.