The Solution of Nonlinear Elliptic Dirichlet Problems on Rectangles by Almost Globally Convergent Interval Methods

Abstract

The use of interval arithmetic offers the possibility to develop methods for the solution of systems of nonlinear equations that converge to the solution under relatively weak conditions, provided an initial inclusion is known. In the present paper we describe methods for discretizations of nonlinear elliptic equations, in particular for Dirichlet problems of the type $(au_x )_x + (bu_y )_y = f(x,y,u)$ where $a \equiv a(x)$ and $b = b(y)$ or $a \equiv a(x,y)$, $b \equiv b(x,y)$. The interval arithmetic enables us to combine a Newton-like interval method with the concept of fast direct solvers. The convergence to the solution can be ensured by auxiliary methods or by monotonicity arguments. We compare some of the possible variants with the generalized CG method and the nonlinear block SOR method.