An efficient modular algorithm for coupled nonlinear systems

Abstract

We present an efficient modular algorithm for solving the coupled system G(u,t)=0 and N(u,t)=0 where u∈Rⁿ, t∈R^m, G:Rⁿ×R^m→Rⁿ and N: Rⁿ×R^m→R^m. The algorithm is modular in the sense that it only makes use of the basic iteration S of a general solver for the equation G(u,t)=0 with t fixed. It is therefore well-suited for problems for which such a solver already exists or can be implemented more efficiently than a solver for the coupled system. Local convergence results are given. Basically, if S is sufficiently contractive for G, then convergence for the coupled system is guaranteed. The algorithm is applied to two applications: (1) numerical continuation methods and (2) constrained optimization. Numerical results are given for the case where G represents a nonlinear elliptic operator. Three choices of S are considered: Newton's method, a two-level nonlinear multi-grid solver and a supported Picard iteration.

This work was supported in part by the Department of Energy under contract DE-AC02-81ER10996, and by the Army Research Office under contract DAAG-83-0177.