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. Transonic flow simulation by finite elements: Mach distribution. Mach at infinity: 0.85; angle of attack: 1°. (Avions Marcel Dassault-Breguet Aviation, Falcon 50). Bibliography N.B. The following abbreviations for references have been used in the text: G.L.T. for Glowinski R., Lions J. L., Tremolieres R. B.G.4.P. for Bristeau M. O., Glowinski R., Periaux J., Perrier P., Pironneau O., Poirier G. AASEN J. O. [1] On the reduction of a symmetric matrix to tridiagonal form. BIT 11, 233-242 (1971) ADAMS R. A. [1] Sobolev Spaces (Academic, New York 1975) AGMON S., DOUOLIS A., NIRENBERG L. [1] Estimates near the boundary for solution of elliptic partial differential equations satisfying general boundary conditions (I). Commun. Pure Appl. Math. 12, 623-727 (1959) AMANN H. [1] Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18 (4), 620-709 (1976) AMARA M., JOLY P., THOMAS J. M. [1] A mixed finite element method for solving transonic flow equations. Comput. Methods Appl. Mech. Ena. 39, 1-19 (1983) ARGYRIS J. H., DUNNE P. C. [1] "The Finite Element Method Applied to Fluid Mechanics", in Computational Methods and Problems in Aeronautical Fluid Dynamics, B. L. Hewitt, C. R. Illingworth, R. C. Lock, K. W. Mangier, J. H. McDonnell, C. Richards, F. Walkden (Academic, London 1976) pp. 158-197 AUBIN J. P. [1] Mathematical Methods of Game and Economic Theory (North-Holland, Amsterdam 1979) [2] Approximation of Elliptic Boundary Value Problems (Wiley-Interscience, New York 1972) AUSLENDER A. [1] Methodes numeriques pour la decomposition et la minimisation de fonctions non differ- entiables. Numer. Math. 18, 213-223 (1972) AXELSSON O. [1] A class of iterative methods for finite element equations. Comput. Methods Appl. Mech. Eng. 19, 123-138(1976) Aziz A. K., BABUSKA I. [1] "Survey Lectures on the Mathematical Foundations of the Finite Element Method", The Mathematical Foundations of the Finite Element Method with Applications to Partial Differ- ential Equations, ed. by A. K. Aziz (Academic, New York 1972) pp. 3-359 BAIOCCHI C. [1] Sur un probleme a frontiere libre traduisant Ie filtrage de liquides a travers des milieux poreux. C. R. Acad. Sci. Ser. A 273, 1215-1217 (1971) BAIOCCHI C, CAPELO A. [1] Dissequazioni Variazionali e Quasi-variazionali. Applicazioni a problemi di frontier a libera. Quaderni 4 and 7 dell' Unione Mathematica Italiana. (Pitagora Editrice, Bologna 1978) 436 Bibliography BARTELS R., DANIEL J. W. [1] "A Conjugate Gradient Approach to Nonlinear Boundary Value Problems in Irregular Regions," in Conference on the Numerical Solution of Differential Equations, Dundee 197S. Lecture Notes in Mathematics, Vol. 363, ed. by G. A. Watson (Springer, Berlin, Heidelberg, New York 1974) p. 7-77 BARWELL W., GEORGE A. [1] A comparison of algorithms for solving symmetric indefinite systems of linear equations. ACM Trans. Math. Software,! (3) 242-251 (1976) BATHE K. J., WILSON E. L. [1] Numerical Methods in Finite Element Analysis (Prentice-Hall, Englewood Cliffs, NJ 1976) BAUER F., GARABEDIAN P., KORN D. [1] A Theory of Supercritical Wing Sections, with Computer Programs and Examples, Lectures Notes in Economics and Mathematical Systems, Vol. 66 (Springer, Berlin, Heidelberg, New York 1972) BAUER F., GARABEDIAN P., KORN D., JAMESON A. [1] Supercritical Wing Sections, Lecture Notes in Economics and Mathematical Systems, Vol. 108 (Springer, Berlin, Heidelberg, New York 1975) [2] Supercritical Wing Sections, Lecture Notes in Economics and Mathematical Systems, Vol. 150 (Springer Berlin, Heidelberg, New York 1977) BEALE J. T., MAJDA A. [1] Rates of convergence for viscous splitting of the Navier-Stokes equations. Math. Comput. 37,243-260(1981) BEAM R. M., WARMING R. F. [1] An implicit factored scheme for the compressible Navier-Stokes equations. AIAA 16, 393-402 (1978) [2] Alternating Direction implicit methods for parabolic equations with a mixed derivative. SIAMJ. Sci. Stat. Comput. 1 (1) 131-159 (1980) BEGIS D. [1] "Analyse numerique de l'ecoulement d'un fluide de Bingham"; These de 3eme cycle, Universite Pierre et Marie Curie, Paris (1972) [2] " Etude numerique de l'ecoulement d'un fluide visco-plastique de Bingham par une methode de lagrangien augmente." lRIA-Laboria Rpt. 355 (1979) BEGIS D., GLOWINSKI R. [1] "Application des methodes de lagrangien augmente a la simulation numerique d'ecoule- ments bi-dimensionnels de fluides visco-plastiques incompressibles ", in Methodes de lagran- gien augmente. Application a la resolution numerique de problemes aux limites, ed. by M. Fortin, R. Glowinski (Dunod-Bordas, Paris 1982) pp. 219-240 BENQUE J. P., IBLER B., KERAMSI A., LABADIE G. [1] "A Finite Element Method for Navier-Stokes Equations," in Proceedings of the Third Inter- national Conference on Finite Elements in Flow Problems, Banff, Alberta, Canada, 10-13 June 1980, ed. by D. H. Norrie, Vol. 1, pp. 110-120 BENSOUSSAN A., LIONS J. L. [1] Controle Impulsionnel et Inequations Quasi-Variationnelles (Dunod-Bordas, Paris 1982) BERCOVIER M., ENGELMAN M. [1] A finite element method for the numerical solution of viscous incompressible flows. J. Comput. Phys. 30, 181-201 (1979) BERCOVIER M., PIRONNEAU O. [1] Error estimates for finite element method solution of the Stokes problem in the primitive variables. Numer. Math. 33, 211-224 (1979) BERGER A. E. [1] The truncation method for the solution of a class of variational inequalities. Rev. Fr. Autom. Inf. Rech. Oper. 10 (3) 29-42 (1976) BERNADOU M., BOISSERIE J. M. [1] The Finite Element Method in Thin Shell Theory: Application to Arch Dam Simulations (Birkhauser, Boston 1982) Bibliography 437 BERS L. [1] Mathematical Aspects of Subsonic and Transonic Gas Dynamics (Chapman and Hall, London 1958) BLANC M., RICHMOND A. D. [1] The ionospheric disturbance dynamo. J. Geophys. Res. 85 A (4), 1669-1686, (1980) BOURGAT J. F., DUMAY J. M., GLOWINSKI R. [1] Large displacement calculations of flexible pipelines by finite elements and nonlinear pro- gramming methods. SIAM J. Sc. Stat. Comput. 1, 34-81 (1980) BOURGAT J. F., DUVAUT G. [1] Numerical analysis of flows with or without wake past a symmetric two-dimensional profile, with or without incidence. Int. J. Numer. Methods Eng. 11, 975-993 (1977) BOURGAT J. F., GLOWINSKI R., LE TALLEC P. [1] "Application des methodes de lagrangien augmente a la resolution de problemes d'elast- icite non lineaire finie," in Methodes de lagrangien augmente. Application a la resolution numerique de problemes aux limites, ed. by M. Fortin, R. Glowinski (Dunod-Bordas, Paris 1982) pp. 241-278 BREDIF M. [1] " Resolution des equations de Navier-Stokes par elements finis mixtes"; These de 3eme cycle, Universite Pierre et Marie Curie, Paris (1980) BRENT R. [1] Algorithms for Minimization Without Derivatives (Prentice-Hall, Englewood Cliffs, NJ 1973) BREZIS H. [1] "A New Method in the Study of Subsonic Flows," in Partial Differential Equations and Related Topics, ed. by J. Goldstein, Lecture Notes in Mathematics, Vol. 446 (Springer, Berlin, Heidelberg, New York 1975) pp. 50-60 [2] Multiplicateur de Lagrange en torsion elasto-plastique. Arch. Ration. Mech. Anal. 49, 30-40 (1972) [3] Problemes unilateraux. /. Math. Pures Appl. 9, (72), 1 -168 (1971). [4] "Monotonicity in Hilbert Spaces and Some Applications to Nonlinear Partial Differential Equations," in Contributions to Nonlinear Functional Analysis, ed. by E. Zarantonello (Academic, New York 1971) pp. 101-116. [5] Operaleurs maximaux monotones et semi-groupes de contraction dans les espaces de Hilbert (North-Holland, Amsterdam 1973) BREZIS H., CRANDALL M., PAZY A. [1] Perturbation of nonlinear monotone sets in Banach spaces. Commun. Pure Appl. Math. 23, 153-180(1970) BREZIS H., SIBONY M. [1] Equivalence de deux inequations variationnelles et applications. Arch. Ration. Mech. Anal. 41, 254-265 (1971) [2] Methodes d'approximation et d'iteration pour les operateurs monotones. Arch. Ration. Mech. Anal. 28, 59-82 (1968) BREZIS H., STAMPACCHIA G. [1] The hodograph method in fluid dynamics in the light of variational inequalities. Arch. Ration. Mech. Anal. 61, 1-18 (1976) [2] Sur la regularity de la solution d'inequations elliptiques. Bull. Soc. Math. Fr. 96, 153-180 (1968) BREZZI F. [1] " Non-Standard Finite Elements for Fourth Order Elliptic Problems," in Energy Methods in Finite Element Analysis, ed. by R. Glowinski, E. Y. Rodin, O. C. Zienkiewicz (Wiley, Chichester 1979) pp. 193-211 BREZZI F., HAGER W. W., RAVIART P.A. [1] Error estimates for the finite element solution of variational inequalities; Part I: Primal Theory. Numer. Math. 28, 431^43 (1977) [2] Error estimates for the finite element solution of variational inequalities; Part II: Mixed methods. Numer. Math. 31, 1-16 (1978) [...]... Method for Nonlinear Constraints in Minimization Problems, " in Optimization, ed by R Fletcher (Academic, London 1969) pp 283-298 Restart procedure for the conjugate gradient method Math Program 12, 148-162 (1977) " A Hybrid Method for Nonlinear Equations," in Numerical Methods for Nonlinear Algebraic Equations, ed by Ph Rabinowitz (Gordon and Breach, London 1970) pp 87-114 " A Fortran Subroutine for Solving... Solution of Nonlinear Differential Equations, ed by D Greenspan (Wiley, New York 1966) pp 265-296 Approximate solutions and error bounds for quasilinear elliptic boundary value problems SIAM J Numer Anal 7, 80-103 (1970) SCHECHTER S [1] [2] [3] Iteration methods for nonlinear problems Trans Am Math Soc 104, 601- 612 (1962) " Minimization of Convex Functions by Relaxation," in Integer and Nonlinear Programming,... equations by linear programming SIAM Rev 20, 139-167(1978) CHORIN A J [1] [2] [3] [4] A numerical method for solving incompressible viscous flow problems / Comput Phys 2, 12- 26(1967) " Numerical solution of incompressible flow problems, " in Studies in Numerical Analysis, 2; Numerical Solution of Nonlinear Problems (Symposium, SIAM, Philadelphia, Pa., 1968) (Soc Indust Appli Math., Philadelphia, Pa., 1970),... Element Method for the Biharmonic Equation," in Mathematical Aspects of Finite Elements in Partial Differential Equations, ed by C de Boor (Academic, New York 1974) pp 125 -145 ClARLET P G., SCHULTZ M H., VARGA R S [1] Numerical methods of high order accuracy for nonlinear boundary value problems V Monotone operator theory Numer Math 13, 51-77 (1969) ClARLET P G., WAGSHAL C [1] Multipoint Taylor formulas... linearized stability Arch Ration Mech Anal 52, 161-180 (1973) Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems Arch Ration Mech Anal 58, 207-218 (1975) CRISFIELD M A [1] "Finite Element Analysis for Combined Material and Geometric Nonlinearities," in Nonlinear Finite Element Analysis in Structural Mechanics, ed by W Wunderlich, E Stein, K J Bathe... algorithm for minimizing a function of many variables Math Program 23, 20-33 (1982) 450 Bibliography OPIAL Z [1] Weak convergence of the successive approximation for non expansive mappings in Banach spaces Bull Am Math Soc 73, 591-597 (1967) ORTEGA J., RHEINBOLDT W C [1] Iterative Solution of Nonlinear Equations in Several Variables (Academic, New York 1970) OSBORNE M R., WATSON G A [1] " Nonlinear. .. BREZZI F., RAPPAZ J., RAVIART P A [1] [2] [3] Finite dimensional approximation of nonlinear problems, Part I: Branches of nonsingular solutions Numer Math 36, 1-25 (1980) Finite dimensional approximation of nonlinear problems, Part II: Limit points Numer Math 37, 1-28 (1981) Finite dimensional approximation of nonlinear problems, Part III: Simple bifurcation points Numer Math 38, 1-30 (1981) BREZZI F.,... Approximation of Navier-Stokes Equations for Incompressible Viscous Fluids Iterative Methods of Solution," in Approximation Methods for Navier-Stokes Problems, ed by R Rautmann, Lecture Notes in Mathematics, Vol 771 (Springer, Berlin Heidelberg, New York 1980) pp.78 -128 BRISTEAU M O., GLOWINSKI R., PERIAUX J., PERRIER P., PIRONNEAU O [1] On the numerical solution of nonlinear problems in fluid dynamics by least... positive-definite matrix A' the transpose matrix of A CO a relaxation parameter V x v = curl of v a velocity potential . J. [1] A numerical method for solving incompressible viscous flow problems. /. Comput. Phys. 2, 12- 26(1967) [2] " Numerical solution of incompressible flow problems, " in Studies in Numerical. 1980) pp.78 -128 BRISTEAU M. O., GLOWINSKI R., PERIAUX J., PERRIER P., PIRONNEAU O. [1] On the numerical solution of nonlinear problems in fluid dynamics by least squares and finite element methods. . Approximation of Navier-Stokes Equations for Incompressible Viscous Fluids. Iterative Methods of Solution," in Approximation Methods for Navier-Stokes Problems, ed. by R. Rautmann, Lecture