×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ashcraft, C.; Grimes, R., On vectorizing incomplete factorization and SSOR preconditioners, SIAM J. sci. statist. comput., 9, 122-151, (1988) · Zbl 0641.65028
[2] Averick, B., Solution of nonlinear Poisson-type equations on vector computers, () · Zbl 0755.65101
[3] Chan, T.; Kuo, C.; Tong, C., Parallel elliptic preconditioners: Fourier analysis and performance on the connection machine, Comput. phys. comm., 53, 237-252, (1989) · Zbl 0798.65037
[4] Dembo, R.; Eisenstat, S.; Steihaug, T., Inexact Newton methods, SIAM J. numer. anal., 19, 400-408, (1982) · Zbl 0478.65030
[5] Duff, I.; Meurant, G., The effect of ordering on preconditioned conjugate gradients, Bit, 29, 635-657, (1989) · Zbl 0687.65037
[6] Ortega, J., Introduction to parallel and vector solution of linear systems, (1988), Plenum New York
[7] Ortega, J.; Rheinboldt, W., Iterative solution of nonlinear equations in several variables, (1970), Academic Press New York · Zbl 0241.65046
[8] Saad, Y.; Schultz, M., GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. sci. statist. comput., 7, 856-869, (1986) · Zbl 0599.65018
[9] van der Vorst, H., ICCG and related methods for 3D problems on vector computers, Comput. phys. comm., 53, 223-235, (1989) · Zbl 0798.65036
[10] T. Zang, Private communication (1989).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.