Newton's iteration with a conjugate gradient based decomposition method for an elliptic PDE with a nonlinear boundary condition

Jonas Koko

International Journal of Applied Mathematics and Computer Science (2004)

  • Volume: 14, Issue: 1, page 13-18
  • ISSN: 1641-876X

Abstract

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Newton's iteration is studied for the numerical solution of an elliptic PDE with nonlinear boundary conditions. At each iteration of Newton's method, a conjugate gradient based decomposition method is applied to the matrix of the linearized system. The decomposition is such that all the remaining linear systems have the same constant matrix. Numerical results confirm the savings with respect to the computational cost, compared with the classical Newton method with factorization at each step.

How to cite

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Koko, Jonas. "Newton's iteration with a conjugate gradient based decomposition method for an elliptic PDE with a nonlinear boundary condition." International Journal of Applied Mathematics and Computer Science 14.1 (2004): 13-18. <http://eudml.org/doc/207673>.

@article{Koko2004,
abstract = {Newton's iteration is studied for the numerical solution of an elliptic PDE with nonlinear boundary conditions. At each iteration of Newton's method, a conjugate gradient based decomposition method is applied to the matrix of the linearized system. The decomposition is such that all the remaining linear systems have the same constant matrix. Numerical results confirm the savings with respect to the computational cost, compared with the classical Newton method with factorization at each step.},
author = {Koko, Jonas},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {Newton's method; conjugate gradient method; nonlinear PDE},
language = {eng},
number = {1},
pages = {13-18},
title = {Newton's iteration with a conjugate gradient based decomposition method for an elliptic PDE with a nonlinear boundary condition},
url = {http://eudml.org/doc/207673},
volume = {14},
year = {2004},
}

TY - JOUR
AU - Koko, Jonas
TI - Newton's iteration with a conjugate gradient based decomposition method for an elliptic PDE with a nonlinear boundary condition
JO - International Journal of Applied Mathematics and Computer Science
PY - 2004
VL - 14
IS - 1
SP - 13
EP - 18
AB - Newton's iteration is studied for the numerical solution of an elliptic PDE with nonlinear boundary conditions. At each iteration of Newton's method, a conjugate gradient based decomposition method is applied to the matrix of the linearized system. The decomposition is such that all the remaining linear systems have the same constant matrix. Numerical results confirm the savings with respect to the computational cost, compared with the classical Newton method with factorization at each step.
LA - eng
KW - Newton's method; conjugate gradient method; nonlinear PDE
UR - http://eudml.org/doc/207673
ER -

References

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  8. Golub G.H., Murray W. and Saunders M.A. (1974): Methods for modifying matrix factorizations - Math. Comp., Vol.28, No.126, pp.505-535. Zbl0289.65021
  9. Hughes J.T., Ferency R.M. and Halquist J.O. (1987): Large-scale vectorized implicit calculations in solid mechanics on a Cray X-MP/48 utilizing EBE preconditioned conjugate gradient. - Comput. Meth. Appl. Mech. Eng., Vol.61, pp.215-248. 
  10. Saad Y. (1990): SPARSKIT: A basic tool kit for sparse matrix computation. - Tech. Rep. CSRD TR 1029, University of Illinois, Urbana, IL. 
  11. Sonnenveld P., Wesseling P. and De Zeeuw P.M. (1985): Multigrid and conjugate gradient methods as convergence acceleration technique, In: Multigrid Meth. Integr. Diff. pp.117-167, Clarendon Press. 

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