TY  - BOOK
AU  - Neuberger, J. W. 
TI  - Sobolev gradients and differential equations: - 2nd Ed
T2  - Lecture Notes in Mathematics 
SN  - 9783642040405
AV  - QA3 
PY  - 1997///]
CY  - New York
PB  - Springer
N1  - 1. Several Gradients
2. Comparison of Two Gradients
3. Continuous Steepest Descent in Hilbert Space: Linear Case
4. Continuous Steepest Descent in Hilbert Space: Nonlinear Case
5. Orthogonal Projections, Adjoints and Laplacians
6. Ordinary Differential Equations and Sobolev Gradients
7. Convexity and Gradient Inequalities
8. Boundary and Supplementary Conditions
9. Continuous Newton’s Method
10. More About Finite Differences
11. Sobolev Gradients for Variational Problems
12. An Introduction to Sobolev Gradients in Non-Inner Product Spaces
13. Singularities and a Simple Ginzburg-Landau Functional
14. The Superconductivity Equations of Ginzburg-Landau
15. Tricomi Equation: A Case Study
16. Minimal Surfaces
17. Flow Problems and Non-Inner Product Sobolev Spaces
18. An Alternate Approach to Time-dependent PDEs
19. Foliations and Supplementary Conditions I
20. Foliations and Supplementary Conditions II
21. Some Related Iterative Methods for Differential Equations
22. An Analytic Iteration Method
23. Steepest Descent for Conservation Equations
24. Code for an Ordinary Differential Equation
25. Geometric Curve Modeling with Sobolev Gradients
26. Numerical Differentiation, Sobolev Gradients
27. Steepest Descent and Newton’s Method and Elliptic PDE
28. Ginzburg-Landau Separation Problems
29. Numerical Preconditioning Methods for Elliptic PDEs
30. More Results on Sobolev Gradient Problems
31. Notes and Suggestions for Future Work
N2  - A Sobolev gradient of a real-valued functional on a Hilbert space is a gradient of that functional taken relative to an 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. For discrete versions of partial differential equations, corresponding Sobolev gradients are seen to be vastly more efficient than ordinary gradients. In fact, descent methods with these gradients generally scale linearly with the number of grid points, in sharp contrast with the use of ordinary gradients. Aside from the first edition of this work, this is the only known account of Sobolev gradients in book form.
ER  -