Numerical verification methods for solutions of semilinear elliptic boundary value problems

This article describes a survey on numerical verification methods for second-order semilinear elliptic boundary value problems introduced by authors and their colleagues. Here "numerical verification" means a computer-assisted numerical method for proving the existence of a solution in a close and explicit neighborhood of an approximate solution. Three kinds of methods based on the infinite dimensional fixed-point theorems using Newton-like operator will be presented. In each verification method, a projection into a finite dimensional subspace and constructive error estimates of the projection play an important and essential role. It is shown that these methods are really useful for actual problems by illustrating numerical examples.