Sobolev gradients and differential equations / / J. W. Neuberger

• Record Nr.: UNINA9910146292203321
• Autore: Neuberger J. W (John W.), <1934->
• Pubbl/distr/stampa: Berlin, Germany ; ; New York, New York : , : Springer, , [1997]; ©1997
• ISBN: 3-540-69594-X
• Edizione: [1st ed. 1997.]
• Descrizione fisica: 1 online resource (VIII, 152 p.)
• Collana: Lecture Notes in Mathematics, , 0075-8434 ; ; 1670
• Classificazione: 65N30; 35A15
• Disciplina: 515/.353
• Soggetti: Differential equations - Numerical solutions; Sobolev gradients
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Note generali: Bibliographic Level Mode of Issuance: Monograph
• Nota di bibliografia: Includes bibliographical references (pages [145]-149) and index.
• Nota di contenuto: 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.
• Sommario/riassunto: 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.