A generalization of Uzawa’s algorithm for the solution of the Navier- Stokes equations. (English) Zbl 0592.76040

Fortin, M.; Fortin, A.

Commun. Appl. Numer. Methods 1, 205-208 (1985).

Usawa’s algorithm [K. J. Arrow, L. Hurwicz and H. Uzawa, Studies in nonlinear programming (1958; Zbl 0091.160)] provides an efficient method for solving the divergence-free Stokes problem. The Newton-Raphson scheme is very popular for the solution of the nonlinear Navier-Stokes equations. We propose here a new method that combines these two algorithms and converges to a divergence-free solution of the nonlinear Navier-Stokes equations.

MSC:

76D05	Navier-Stokes equations for incompressible viscous fluids
35Q30	Navier-Stokes equations
76M99	Basic methods in fluid mechanics

Keywords:

homogeneous boundary conditions; saddle-points of the Lagrangian; functional; divergence-free Stokes problem; Newton-Raphson scheme; nonlinear Navier-Stokes equations; divergence-free solution

Citations:

Zbl 0091.160

References:

[1]	and , Studies in Non-linear Programming, Stanford Univ. Press, California, 1958.
[2]	and , Résolution Numérique de Problèmes aux limites par des Méthodes de Lagrangien Augmenté, Dunod, Paris, 1983.
[3]	and , ’Finite element approximation of the Navier-Stokes equations’, in Lecture Notes in Mathematics, Springer Verlag, Berlin, 1979.
[4]	Fortin, Int. j. numer. methods fluids 5 (1985)