Computer-assisted enclosure methods for elliptic differential equations Original Research Article

We report on methods for computing enclosures of solutions of second-order nonlinear elliptic boundary value problems, simultaneously proving the existence of a solution in the enclosing set. The old-fashioned ‘monotonicity methodsʹ are well suited for this task, but only for a restricted class of problems. Therefore, we propose a new approach which is based on a suitable fixed-point formulation of the problem and uses, as an essential ingredient, norm bounds for the inverse of the linearization of the given problem at some approximate solution ω which is computed numerically. These norm bounds are obtained via eigenvalue enclosures. We also give a brief description of an alternative method proposed by M.T. Nakao.