03645nam 2200637 450 991014629220332120220420154900.03-540-69594-X10.1007/BFb0092831(CKB)1000000000437343(SSID)ssj0000326692(PQKBManifestationID)12069599(PQKBTitleCode)TC0000326692(PQKBWorkID)10296698(PQKB)10963532(DE-He213)978-3-540-69594-3(MiAaPQ)EBC5577146(MiAaPQ)EBC6691771(Au-PeEL)EBL5577146(OCoLC)1066180869(Au-PeEL)EBL6691771(PPN)155195549(EXLCZ)99100000000043734320220420d1997 uy 0engurnn#008mamaatxtccrSobolev gradients and differential equations /J. W. Neuberger1st ed. 1997.Berlin, Germany ;New York, New York :Springer,[1997]©19971 online resource (VIII, 152 p.)Lecture Notes in Mathematics,0075-8434 ;1670Bibliographic Level Mode of Issuance: Monograph3-540-63537-8 Includes bibliographical references (pages [145]-149) and index.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.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.Lecture Notes in Mathematics,0075-8434 ;1670Differential equationsNumerical solutionsSobolev gradientsDifferential equationsNumerical solutions.Sobolev gradients.515/.35365N30msc35A15mscNeuberger J. W(John W.),1934-61864MiAaPQMiAaPQMiAaPQBOOK9910146292203321Sobolev gradients and differential equations374787UNINA