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Solution of nonlinear Poisson-type equations. (English) Zbl 0755.65101
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.

##### MSC:
 65N06 Finite difference methods for boundary value problems involving PDEs 65N22 Numerical solution of discretized equations for boundary value problems involving PDEs 65H10 Numerical computation of solutions to systems of equations 35J65 Nonlinear boundary value problems for linear elliptic equations
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