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Tham khảo tài liệu 'numerical_methods for nonlinear variable problems episode 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 6. Applications 187 Exercise 6.1. Prove 6.28 6.31 . We refer to Glowinski and Marrocco 5 for a detailed analysis including error estimates of a finite element approximation of 6.26 6.27 see also Ciarlet 2 . From our numerical experiments it appears that solving 6.26 6.27 if s is close to 1 say 1 s 1.3 or large say s 5 is a very difficult task if one uses standard iterative methods to our knowledge the only very efficient methods are ALG 1 and ALG 2 or closely related algorithms see Glowinski and Marrocco loc. cit. for more details . The augmented Lagrangian r to be used for solving 6.26 6.27 is defined by If Ỵ f f q ụ - I q Is dx f v 7 I Vt q 2 dx p Vi q dx. s Jq 2 Jii Jn 6.32 Solution of 6.26 6.27 by ALG 1. From 3.2 - 3.4 6.32 it follows that when applying ALG 1 to 6.26 6.27 we obtain Ằ e Ls O N- 6.33 then for n 0 rAm f V 2 rV pn in Q __ 6.34 u r 0 p s 2p rpn rVun 2 6.35 2 1 2 p Vm - p . 6.36 The nonlinear system 6.34 6.35 can be solved by the block-relaxation method of Sec. 4.3 and we observe that if u and 2 are known or estimated in 6.35 the computation of p is an easy task since p is solution of the singlevariable nonlinear equation p s-1 r p rVu 2 6.37 which can easily be solved by various methods once I pn I is known we obtain pn by solving a trivial linear equation in L G iV . Solution of 6.26 6.27 by ALG 2. We have to replace 6.33 by p Ả eHxH 6.38 and 6.34 by rAw V 2 rV p 1 __ 6.39 w r 0. 188 VI Decomposition-Coordination Methods by Augmented Lagrangian Applications Remark 6.1 still applies to 6.26 6.27 and since G is linear we can take 0 p p 2r if we are using ALG 2. For more details and comparisons with other methods see Glowinski and Marrocco 5 7 and Fortin Glowinski and Marrocco 1 . Remark 6.2. ALG 1 and ALG 2 have also been successfully applied to the iterative solution of magneto-static problems see Glowinski and Marrocco 6 . They have also been applied by Fortin Glowinski and Marrocco 1 to the solution of the subsonic flow problem described in
