Solution Manifolds and Submanifolds of Parametrized Equations and Their Discretization Errors.

摘要

The paper concerns solution manifolds of nonlinear parameter-dependent equations (1) F(u,lambda) = y sub o involving a Fredholm operator F between (infinite-dimensional) Banach spaces X = Z x lambda and Y, and a finite-dimensional parameter space lambda. Differential-geometric ideas are used to discuss the connection between augmented equations and certain one-dimensional submanifolds produced by numerical path-tracing procedures. Then, for arbitrary (finite) dimension of lambda, estimates of the error between the solution manifold of (1) and its discretizations are developed. These estimates are shown to be applicable to rather general nonlinear boundary-value problems for partial differential equations. (Author)