Newton's method for the Navier-Stokes equations with finite-element initial guess of stokes equations

K e y w o r d s N a v i e r S t o k e s equations, Stokes equations, Convergence, Finite element method, Newton's method. 1. I N T R O D U C T I O N In the course of so lv ing non l inear equa t ions like Navie r -S tokes equa t ions , one m a y e m p l o y N e w t o n and Newtonl ike m e t h o d s combin ing wi th o the r m e t h o d s (see, for example , [1-3], etc.) . I t is well known t h a t i f t he in i t ia l guess is chosen nea rby the exac t so lu t ion of t h e g iven non l inea r different ial equa t ions the q u a d r a t i c convergence is gua ran t eed (see [4]). I t is known t h a t t he so lu t ion of Stokes equa t ions m a y be chosen as the ini t ia l guess for N e w t o n ' s m e t h o d for Nav ie r -S tokes equa t ions This work was supported by Korea Research Foundation Grant (KRF-2002-070-C00014). 0898-1221/06/$ see front matter (~) 2006 Elsevier Ltd. All rights reserved. Typeset by .A~tS-TF_X doi: 10.1016/j.camwa. 2006.03.007 806 s.D. KIM et al. which we consider now with zero boundary condition for the ve loc i t y u = ( u l , u 2 ) t and the mean-zero condition for the pres sure p as follows: ~ A u + (uV)u + Vp = f, in f~, V u = 0, in t2, u = 0, on 0a , (1.1)