Solution of nonlinear Poisson-type equations (Q1181546)

The boundary value problem in the unit cube in \(\mathbb{R}^ 3\) for the Poisson-type equation \(\nabla\cdot [K(u)\nabla u]=f\) is reduced by means of a finite-difference procedure to a discrete system of the form \(F(u)\equiv A(u)u-b(u)=0\), where \(A(u)\) is a symmetric positive-definite matrix for all \(u\). For the Newton method one has to solve the linear systems \(F'(u^ k)\delta^{k+1}=-F(u^ k)\), \(k=0,1,2,\dots\), where \(F'(u^ k)\) is the in general not symmetric Jacobian matrix and \(\delta^{k+1}\) is the Newton correction vector. The idea of the paper is to avoid conjugate gradient methods and to solve approximately the linear systems \(A(u^ k)\delta^{k+1}=-F(u^ k)\), \(k=0,1,2,\dots\) at each Newton step. It is shown that as the problem size increases, \(A(u)\) becomes a better approximation to the Jacobian matrix, so that the convergence properties of Newton's method should be more closely attained. The numerical results obtained on a SUN 3/60 and CRAY 2 are also presented.