an:01195675
Zbl 0908.65060
Mahavier, W. T.
A numerical method utilizing weighted Sobolev descent to solve singular differential equations
EN
Nonlinear World 4, No. 4, 435-455 (1997).
00048124
1997
j
65L05 34C05 34A34 34B15 65L10 65L12
finite differences; preconditioning; singular differential equations; steepest descent; weighted Sobolev gradients; least-squares problem
Author's abstract: A numerical method is developed for solving singular differential equations using steepest descent based on weighted Sobolev gradients. The differential equation is cast as a least-squares problem yielding a functional representing the equation. A weighted Sobolev space is chosen which depends on both the functional and the boundary conditions, thus gradients associated with the functional take into account the singularity and the boundary conditions for the given equation. Results are presented for a variety of first- and second-order problems, including linear constrained, unconstrained, and partially constrained first-order problems, a nonlinear first-order problem with irregular singularity, and two singular second-order variational problems. Significant improvements are obtained by computing based on weighted Sobolev gradients rather than computing based on unweighted Sobolev gradients.
Z.Jackiewicz (Tempe)