An Interval Arithmetic Approach for the Construction of an Almost Globally Convergent Method for the Solution of the Nonlinear Poisson Equation on the Unit Square

Abstract

The discretization of the nonlinear Poisson equation on the unit square with Dirichlet boundary conditions leads to very large systems of nonlinear equations for small mesh sizes. The use of interval arithmetic enables us to develop a Newton-like method with an interval “fast Poisson solver.” This method converges to the solution of the discretized problem provided an initial inclusion vector is known. The latter is easy to compute. We therefore speak of almost global convergence. Our method competes very well with known algorithms like the generalized conjugate gradient method and others with regard to global convergence, storage requirement and computation time.