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Tham khảo tài liệu 'numerical_methods for nonlinear variable problems episode 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 6. Time-Dependent Flow of a Bingham Fluid in a Cylindrical Pipe 107 6.3. On the asymptotic behavior of the discrete solution We still assume that f 6 L2 Q . To approximate 6.1 we proceed as follows Assuming that Q is a polygonal domain we use the same approximation with regard to the space variables as in Chapter II Sec. 6 i.e. by means of piecewise linear finite elements see Chapter II Sec. 6 . Hence we have ah uh vh a uh vh Vuh vheVh h Vh VvheVh and from the formula of Chapter II Sec. 7 we can also take uh vh h uh vh V uh vheVh. Then we approximate 6.1 by the implicit scheme 5.2 and obtain Uh A Uh vh - Ã 1Ì f Vu 1 V i - u h 1 dx af ph ỵ zxr - gj unh i fh vh - unh l YvheVh unh leVh n 0 1 2 . u uOh. 6.14 We assume that uOh e vh V h and lim uOh Mo strongly in L2 Q . 6.15 h- O Similarly we assume that f is approximated by fffi in such a way that Vh can be computed easily and lim fh f strongly in L2 Q . 6.16 h o Theorem 6.3. Let f Pg. If 6.15 and 6.16 hold then if h is sufficiently small we have u 0 for n large enough. Proof. As in the proof of Theorem 6.2 taking vh 0 and vh 2u h 1 in 6.14 we obtain i ------- uJJ 1 j g f I VuJ 112 dx g f I Vuâ 11 dx f fidf dx V n 0 At Jil Jn 2ij 6.17 using the Schwarz inequality in L2 íĩ it follows from 6.17 that l l K1 4K112 .gp - lÁDK1 0 Vn 0. 6.18 Since fh f strongly in L2 ii we have gP fh 0 for h sufficiently small 6.19 108 III On the Approximation of Parabolic Variational Inequalities From 6.18 6.19 it then follows that u h 0 u h 0 for n n0 if h is small enough. 6.20 Assume that Uh 0 V n then 6.18 implies Mii gff - I.ÁI 0 Vn 0. 6.21 We define 7 by yh gf I fh I then yh 0 for h small enough and lim yh 7 gfi I f I. 6.22 From 6.21 it follows that X 1I tM 1 AoM At w -A- V I 0. -0 pj which implies that l i 1 Ẫoụ At u -A-V 6.23 Since yh 0 for h small enough 6.23 is impossible for n large enough. More precisely we shall have u 0 if - - 1 AoMAt - K -H 0 M A v which implies t Log l UXlX l If h is small enough then Uj 0 if n T - 7- 7 Log l
