Computer-assisted enclosure methods for elliptic differential equations

We report on methods for computing enclosures of solutions of second-order nonlinear elliptic boundary value problems, simultaneously proving the exis tence of a solution in the enclosing set. The old-fashioned 'monotonicity m ethods' are well suited for this task, but only for a restricted class of p roblems. 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 omega which is computed numerically. These norm b ounds are obtained via eigenvalue enclosures. We also give a brief descript ion of an alternative method proposed by M.T Nakao. (C) 2001 Elsevier Scien ce Inc. All rights reserved.