Tham khảo tài liệu 'numerical_methods for nonlinear variable problems episode 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 4. A Third Example of EVI of the First Kind A Simplified Signorini Problem 67 Relation can also be written as q p p pyu p dr 0 V q e A p 0 and p e A which is classically equivalent to p Pa p - PỴÙ . Let us consider Ũ u u p pn p. Since PA is a contraction from and we have IIp 1 Il2 d lip - P IIlad- From it follows that p l2 T - pn 1IL2 D 2p f yunpn dr - p2 yũ Ỉ2 r . Jr Taking V Ũ in and we obtain ữ ũ Ũ f pnyun dr. Jr From and it then follows that IIp IIl2 F - l p 1 IIl2 F p 2 - pIMI2 IIH Ilium- 4-69 If 0 p 2 y 2 we observe that the sequence pn 2 r is decreasing and hence converges. Therefore we have lim IIp IIl2 f IIp 1IIl2 F 0 GO SO that lim ll llnum 0. co Since Ũ u u we have proved the convergence. Similarly we can solve the approximate problems Pl k 1 2 using the discrete version of algorithm - . We shall limit ourselves to k 1 since the extension to k 2 is trivial. Here we use the notations of Sec. . Assume that yh E Ẻ has been ordered. Let yh 68 II Application of the Finite Element Method We approximate A and ỵ by AẪ ỉhl ĩh 0 and h .vh qh ia vh 17ft - L vh - qi 1vh Mi iyi i We can prove that J5 ft has a unique saddle point uh ph where ph is a F. John-Kuhn- Tucker vector for P ft over vlh X Aft and Uft is precisely the solution of Pjft . The discrete analogue of - is then I M PĨ ỉ vh PĨ VvheVl uĩevl p 1 p - Vi p 0. One can prove that if 0 p 1 with Ismail enough then lim zUft Uft where Uft is the solution of P ft . In . 3 Chapter 4 one may find numerical applications of the above iterative methods for piecewise linear and piecewise quadratic approximations of . Exercise . Extend the above considerations to Pfft . 5. An Example of EVI of the Second Kind A Simplified Friction Problem . The continuous problem. Existence and uniqueness results Let Q be a bounded domain of R2 with a smooth boundary r ỔQ. Using the same notations as in Sec. 4 we define
