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.
Reihe
Sprache
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Verlagsgruppe
Zielgruppe
Für höhere Schule und Studium
Für Beruf und Forschung
Produkt-Hinweis
Illustrationen
Maße
Höhe: 23.4 cm
Breite: 15.6 cm
Dicke: 8 mm
Gewicht
ISBN-13
978-3-540-63537-6 (9783540635376)
DOI
Schweitzer Klassifikation
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.
