Numerical_Methods for Nonlinear Variable Problems Episode 12 pps

5 Decomposition Properties of the Continuous and Discrete Stokes Problems of Sec. 4. 427 We observe that the boundary condition on T t is quite formal since p", as an element of L 2 (Q), usually has no trace on T l ; to overcome this difficulty, we shall use a variational formulation of (5.8), namely u n e(H\Q)) N , u" = g 0 onr 0 , and a I u" • v dx + v ( Vu" • Vv dx = \ f • v dx + \ p" V • v dx + \ g x • v dT, •In Jn Jn Jn Jr, V v e (H\n)) N , v = 0 on T o . (5.8)' About the convergence of algorithm (5.7)-(5.9), we have: Proposition 5.1. Suppose that 0<p<2 —. (5.10) We then have, V p° e L 2 (Q), lim {u", p"} = {u, p} strongly in (H^Q.))" x L 2 (Q), (5.11) * + oo where {u, p} is the solution of (5.1), (5.2). PROOF. Define u" and p" by u" = u" - u and p" = p" - p. We clearly have u" e (H\ayf, u" = 0 on T o and a \vT-vdx + v (*Vu"-Vvdx= \f\-ydx, M y e{H l (a)) N , v = 0onr 0 , •>si •'n •'n (5.12) and (since V • u = 0) p»+i =p"- p V-0". (5.13) From (5.13) it follows that \\P"\\h m - \\P- +I \\l m = 2p f P"V • u" dx - p 2 f |V • u"| 2 rfx. (5.14) •>n •'n Now taking v = u" in (5.12), and combining with (5.14), we obtain »\ 2 dx + v f \\u"\ 2 dx)-p 2 f|V-u"| 2 ^. (5.15) Combining (5.15) with relation (5.325) of Chapter VII, Sec. 5.8.7.4.3 (i.e., a f l y l 2 dx + v f I Vv l 2 dx ' v v 428 App. Ill Some Complements on the Navier-Stokes Equations we finally obtain which proves the convergence of u" to u, V p° e L 2 (il), if (5.10) holds (we have to remember that Q bounded implies that \ 1/2 / r r V a \\\ 2 dx + v \\v\ 2 dx) is a norm on {v|v e (H 1 (Q)) N , v = 0 on F o }, equivalent to the (H 1 (n)) N -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 {u", p"} to {u, p} is linear). Remark 5.1. Using the material of Chapter VII, Sec. 5.8.7.4, it is straight- forward to obtain conjugate gradient variants of algorithm (5.7)-(5.9) and also variants derived from an augmented Lagrangian functional reinforcing the incompressibility condition. The same observations hold for the solution of the approximate problem (5.4). Remark 5.2. When using a finite element variant of algorithm (5.7)-(5.9) to solve the approximate problem (5.4), we have to solve, at each iteration, a discrete elliptic system with boundary conditions of the Dirichlet-Neumann type. The solution of such problems has been discussed in Appendix I, Sec. 4. The same observation holds for the conjugate gradient and augmented Lagrangian algorithms mentioned in Remark 5.1 above. 5.4. Solution of (5.1) via (5.3) and (5.5), (5.6) We follow (and generalize) Chapter VII, Sec. 5.7, where the situation F = F o , Fj = 0 was treated. In this section we suppose that j Fl dT > 0. The decomposition properties of the Stokes problem (5.1) follow directly from: Proposition 5.2. LetXeH~ 1/2 (F) and let A: H~ 1/2 (F) -+ H 1I2 (T) be defined by the following cascade of Dirichlet and Dirichlet-Neumann problems. - vAu A = Ap A — \p x in Q, -A\ii x = 0 = V in • u Q, = i in Pi OonF Q, = X 0. on F, V ~dn~ = Oon r, n (= An) (5.16) onT u (5-17) (5.18) 5 Decomposition Properties of the Continuous and Discrete Stokes Problems of Sec. 4. 429 and then AX= - dn (5.19) Then A is an isomorphism from H 1/2 (F) onto H l/2 (T). Moreover, the bilinear form a(-, •) defined by a(X, n) = (AX, /i>, V X, pi e H~ 1/2 (r) (5.20) (where <•, •> denotes the duality pairing between H 1/2 (T) and /J~ 1/2 (F)) is continuous, symmetric, and H~ 1/2 (F)- elliptic. We do not give the proof of Proposition 5.2; let us mention, however, that it is founded on the relation (AX U X 2 ) = a f u Al .u^x + vf Vu Al • Vu, 2 dx, V X,, X 2 e H~ 1/2 (T), •Jn Jn where u Xl , u Az are the solutions of (5.17) corresponding to X = X l and X = X 2 , respectively. Application of Proposition 5.2 to the solution of the Stokes problem (5.1). We define p 0 , u 0 , i// 0 as the solutions of, respectively <xu 0 The — vAu 0 = f - fundamental Ap 0 - \p 0 in -A^ o result is = V- Q, = V- given fin by: Q, -go in £2, Po = onr 0 , "Ao OonF, 3n 8 = 0 on F. (5.21) ! + p 0 n on Fj, (5.22) (5.23) Theorem 5.1. Let {u, p} be the solution of the Stokes problem (5.1). The trace X = p\ x is the unique solution of the linear variational equation V/iEf/-" 2 (r). (5.24) If we compare the above theorem to Theorem 5.7 of Chapter VII, Sec. 5.7.1.2, we observe that this time—due to meas^) > 0—the trace of the pressure is uniquely denned by (5.24). The same decomposition principles can be applied to the discrete Stokes problem (5.5), (5.6); since the resulting methods are trivial variants of the methods discussed in Chapter VII, Sec. 5.7.2, they will not be discussed here any further, except to say that, again, meas(F 1 ) > 0 implies that the linear system (discrete analogue of (5.24)), providing the trace of the discrete pressure p h , has a unique solution. 430 App. Ill Some Complements on the Navier-Stokes Equations 6. Further Comments The methods, for solving the Navier-Stokes equations, discussed in Chapter VII, Sec. 5, and in this appendix have been generalized by Conca [1], [2], in order to treat a large variety of boundary conditions involving the stress tensor 5 = (ff^)^ } defined by a = -p3.j + 2vD u (u), (6.1) where u = {« ; }f = t and D;/u) = ^(duJdXj + 3u/tbc ; ). In this direction, it is quite convenient to use the following equivalent formulation of the Navier- Stokes equations: «« ( -2v£ ^A»+ £ M ,gi + |Uy;.infi, i = l JV, j=l OXj j=i OXj OX t (6.2) V-u = 0 inO. (6.3) We refer to Conca, loc. cit., for further details (see also Engelman, Sani, and Gresho [1] for the practical finite element implementation of various boundary conditions associated with the Navier-Stokes equations). Some Illustrations from an Industrial Application The methods described in Chapter VII have been used for the numerical simulation of the aerodynamical performances of a tri-jet engine AMD/BA Falcon 50. Figure A shows the trace on the aircraft of the three-dimensional finite element mesh used for the computation, and Fig. B shows the correspond- ing Mach distribution (dark: low Mach number, light: high Mach number); the flow is mainly supersonic on the upper part of the wings. 432 An Industrial Application "3 Q 60 E O- c Figure B. 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