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 . Prove . We refer to Glowinski and Marrocco 5 for a detailed analysis including error estimates of a finite element approximation of see also Ciarlet 2 . From our numerical experiments it appears that solving if s is close to 1 say 1 s 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 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 Solution of by ALG 1. From - it follows that when applying ALG 1 to we obtain Ằ e Ls O N- then for n 0 rAm f V 2 rV pn in Q __ u r 0 p s 2p rpn rVun 2 2 1 2 p Vm - p . The nonlinear system can be solved by the block-relaxation method of Sec. and we observe that if u and 2 are known or estimated in the computation of p is an easy task since p is solution of the singlevariable nonlinear equation p s-1 r p rVu 2 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 by ALG 2. We have to replace by p Ả eHxH and by rAw V 2 rV p 1 __ w r 0. 188 VI Decomposition-Coordination Methods by Augmented Lagrangian Applications Remark still applies to 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 . 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