Solution of nonlinear Poisson-type equations.

*(English)*Zbl 0755.65101The 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.

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.

Reviewer: Yu.V.Kostarchuk (Chernigov)

##### 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 |

##### Keywords:

numerical examples; nonlinear Poisson equation; finite-difference methods; Newton’s method; Jacobian matrix; preconditioned conjugate gradient method
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\textit{B. M. Averick} and \textit{J. M. Ortega}, Appl. Numer. Math. 8, No. 6, 443--455 (1991; Zbl 0755.65101)

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