tailieunhanh - Numerical_Methods for Nonlinear Variable Problems Episode 12

Tham khảo tài liệu 'numerical_methods for nonlinear variable problems episode 12', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 5 Decomposition Properties of the Continuous and Discrete Stokes Problems of Sec. 4. 427 We observe that the boundary condition on rx is quite formal since p as an element of L2 Q usually has no trace on Tj to overcome this difficulty we shall use a variational formulation of namely u e H n A u g0 on ro and a I u V dx V f Vu Vv dx I f V dx f p V V dx I gt V dr Jil Jii Jn Jii Jr V V 6 H1 Q ỈV V 0 on r0. About the convergence of algorithm - we have Proposition . Suppose that ũ p 2 . We then have V p e L2 Q lim u pn u p strongly in H Q X L2 il 00 where u p is the solution of . Proof. Define Ũ and p by ũ u u and p p p. We clearly have Ũ e Hfii Ũ 0 on r0 and a f u V dx V f Vũ Vv dx f p V V dx V V e 77 Fl V 0 on r0 hl I l Jn and since V u 0 p 1 p - pV-ũ . From it follows that llp IIỈ2 n - p 1 lit 2p i p V u dx - p2 f V ũ 2 dx. Jq. Now taking V Ũ in and combining with we obtain IIpnII 2 0 - IIp 1IIỈ2 íỉ 2p a f ũ 2 dx V f V0 2 dx - p2 f V-ũ 2dx. Jo Jii Jq Combining with relation of Chapter VII Sec. . J IIV v 22 n a f Iv 2 dx V f I Vv 2 dx V V 6 H íỉlA 428 App. Ill Some Complements on the Navier-Stokes Equations we finally obtain llp ll ii - Iỉp 1IIỈ2 ỉ p 2 dx V which proves the convergence of u to u V p e if holds we have to remember that ÍĨ bounded implies that V - a f v 2 dx V f I Vv I2 dx j Jn Ạỉ is a norm on v v e Q V 0 on r0 equivalent to the H íỊh -norm and this for all a 0 . The proof of the convergence of p to p is left to the reader actually we should prove that the convergence of iT p to u p is linear . Remark . Using the material of Chapter VII Sec. it is straightforward to obtain conjugate gradient variants of algorithm - and also variants derived from an augmented Lagrangian functional reinforcing the incompressibility condition. The same observations hold for the solution of the approximate problem . Remark . When using a finite .