Almost Globally Convergent Interval Methods for Discretizations of Nonlinear Elliptic Partial Differential Equations

Abstract

We introduce interval methods for the numerical solution of nonlinear systems of equations for discretizations of partial differential equations. Similar methods have been discussed in [23]. All these methods go back to an idea of Krawczyk [10] and a similar method presented by Alefeld/Platzoder [3]. They converge under relatively weak conditions to the solution provided an initial inclusion is known. Since such an inclusion can be easily computed for problems of the type we discuss, it can be said that the convergence is (almost) global. The methods of the present paper converge without a particular numerical strategy or auxiliary methods in contrast to the methods in [21], [24]. Also they can be applied to a larger number of problems than those in [21], [24]. Convexity conditions are not needed for the convergence of our methods. The present paper contains a comparison of the iterates produced by the original method of Krawczyk and our methods.