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Tham khảo tài liệu 'numerical_methods for nonlinear variable problems episode 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 5 Numerical Solution of the Navier-Stokes Equations 267 where denotes the duality pairing between H1 2 r and H 1 2 r is continuous symmetric and H ll2 ĩ ữ.-elliptic. We recall that a bilinear form defined over V X V where V is a Hilbert space is F-elliptic if there exists p 0 such that a v v jổ r 2 V V e V. For the proof of Theorem 5.6 see Glowinski Periaux and Pironneau 1 a variant of it concerning the biharmonic problem is avaiable in Glowinski and Pironneau 1 . Exercise 5.4. Prove Theorem 5.6. Application of Theorem 5.6 to the solution of the Stokes problem 5.103 . Let us define p0 u0 ìịjữ as the solutions of respectively Ap0 V f in Q. Po 0 on r 5.109 au0 vAu0 f Vp0 in Q u0 g on T 5.110 Ai 0 V u0 in Q 1 0 0 on T. 5.111 The following result is established in Glowinski Periaux and Pironneau 1 Theorem 5.7. Let u p be a solution of the Stokes problem 5.103 . The trace Ả p r is the unique solution of the linear variational equation E Ả e H l 2 T R AẰ p p V p 6 H 1 2 T R. 5.112 on Exercise 5.5. Prove Theorem 5.7. From the above theorem it follows that the solution of the Stokes problem 5.103 has been reduced to 2N 3 Dirichlet problems21 N 2 to obtain ý0 N 1 to obtain u p once Ấ is known and to the solution of E which is a kind of boundary integral equation. The main difficulty in this approach is the fact that the pseudo-differential operator A is not known explicitely actually the mixed finite element approximation of Sec. 5.3 has been precisely introduced in Glowinski and Pironneau 4 5 to overcome this difficulty extending to the Stokes problem an idea discussed in Glowinski and Pironneau 1 for the biharmonic problem. 5.7.2. The discrete case 5.7.2.1. Introduction formulation of the basic problem. From Secs. 5.4 and 5.6 it follows that efficient solvers for the discrete Stokes problem 21 N 2 if rỉ R2 N 3 ifQ c R3. 268 VII Least-Squares Solution of Nonlinear Problems Find uh fh e Wgh such that V vh ộh 6 WOh we have x Ujj vh dx V Vu VvA dx f0 1 vh dx flh vh V f h dx .
