Reliable Solution for a 1D Quasilinear Elliptic Equation with Uncertain Coefficients

Abstract

The solution of a quasilinear elliptic state equation depends on the coefficient function belonging to an admissible set. The solution is evaluated by a cost functional the value of which is to be maximized over the admissible set, i.e., the reliable (safe) solution is searched for. Due to the nature of the equation, the Kirchhoff transformation can be applied to obtain both the existence of the true state solution and a cost sensitivity formula. In many cases, the latter makes it possible to determine the reliable solution immediately. The problem is approximated by means of the finite element method, and some convergence results are proven. Numerical examples illustrate the theory which can be directly generalized to spatial problems.