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Steepest descent for general systems of linear differential equations in Hilbert space. (English) Zbl 0534.35002
Ordinary differential equations and operators, a Trib. to F. V. Atkinson, Proc. Symp., Dundee/Scotl. 1982, Lect. Notes Math. 1032, 390-406 (1983).
[For the entire collection see Zbl 0514.00013.]
This paper concerns the use of steepest descent to solve boundary value problems for linear systems of partial differential equations. Under very general conditions it is shown that steepest descent will yield a solution if there is one. The effect of choice of norm on the rate of convergence is considered. Two gradients are constructed. One uses a Euclidean norm and the other a discrete version of a Sobolev norm. An example, which is asserted to be typical, indicates a clear advantage of the Sobolev norm.

35A15 Variational methods applied to PDEs
34G10 Linear differential equations in abstract spaces
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
35A35 Theoretical approximation in context of PDEs
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems