Sobolev Gradients and Differential Equations [1 ed.] 3540635378, 9783540635376

A Sobolev gradient of a real-valued functional is a gradient of that functional taken relative to the underlying Sobolev

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English Pages 149 Year 1997

Table of contents :
Several gradients....Pages 1-3
Comparison of two gradients....Pages 5-9
Continuous steepest descent in Hilbert space: Linear case....Pages 11-13
Continuous steepest descent in Hilbert space: Nonlinear case....Pages 15-31
Orthogonal projections, Adjoints and Laplacians....Pages 33-42
Introducing boundary conditions....Pages 43-52
Newton's method in the context of Sobolev gradients....Pages 53-58
Finite difference setting: the inner product case....Pages 59-68
Sobolev gradients for weak solutions: Function space case....Pages 69-73
Sobolev gradients in non-inner product spaces: Introduction....Pages 75-78
The superconductivity equations of Ginzburg-Landau....Pages 79-91
Minimal surfaces....Pages 93-106
Flow problems and non-inner product Sobolev spaces....Pages 107-114
Foliations as a guide to boundary conditions....Pages 115-123
Some related iterative methods for differential equations....Pages 125-133
A related analytic iteration method....Pages 135-138
Steepest descent for conservation equations....Pages 139-140
A sample computer code with notes....Pages 141-143

Sobolev Gradients and Differential Equations [1 ed.]
3540635378, 9783540635376