[...]... Bingham Fluid in a Cylindrical Pipe 10 3 10 4 CHAPTER IV Applications of Elliptic Variational Inequality Methods to the Solution of Some Nonlinear Elliptic Equations 11 0 1 Introduction 2 Theoretical and Numerical Analysis of Some Mildly Nonlinear Elliptic Equations 3 A Subsonic Flow Problem 11 0 11 0 13 4 CHAPTER V Relaxation Methods and Applications 14 0 1 2 3 4 5 14 0 14 0 14 2 15 1 Generalities Some Basic Results... Methods for Convex Functionals: Finite-Dimensional Case Block Relaxation Methods Constrained Minimization of Quadratic Functionals in Hilbert Spaces by Under and Over-Relaxation Methods: Application 6 Solution of Systems of Nonlinear Equations by Relaxation Methods 15 2 16 3 CHAPTER VI Decomposition-Coordination Methods by Augmented Lagrangian: Applications 16 6 1 2 3 4 5 6 7 16 6 16 8 17 0 17 1 17 9 18 3 19 4... 4 .1 by pj(v) and (u, v) + pL(v) — pa(u,v), respectively, we get the solution for (n"p) • Remark 4 .1 FromtheproofofTheorem4 .1 we obtain an algorithm for solving (P 2 ) This algorithm is given by u° e V, arbitrarily given, (4-20) then for n > 0, u" known, we define M" +1 from u" as the solution of (w" +1, v - u" +1) + pj(v) - pj(u" +1) - pa(u", v -un +1) , Vv GK > (un, v - M" + 1 ) + pL(v - M" + 1 ) un +1. .. taking v = u in (5 .12 ), lim | K - u\2 = 0, i.e., the strong convergence • Remark 5.5 Error estimates for the EVI of the first kind can be found in Falk [1] , [2], [3], Mosco and Strang, [1] , Strang [1] , Glowinski, Lions, and Tremolieres (G.L.T.) [1] , [2], [3], Ciarlet [1] , [2], [3], Falk and Mercier [1] , Glowinski [1] , and Brezzi, Hager, and Raviart [1] , [2] But as in many nonlinear problems, the methods... e Kh for any vex, from (P1 (1) we obtain a(uh, uh) < a(uh,rhv) - L(rhv - uh), VUG/ (5 .11 ) Since limh^0 uh = u weakly in V and limft^0 rhv = v strongly in V [by condition (ii)], we obtain (5 .11 ) from (5 .10 ), and after taking the limit, V v e x, we have 0 < a lim inf \\uh - u\\2 < a lim sup||u ft - u\\2 < a(u, v - u) - L(v - u) By density and continuity, (5 .12 ) also holds for VveK; we obtain (5 .12 ) then... Algorithms Convergence of ALG 1 Convergence of ALG 2 Applications General Comments CHAPTER VII Least-Squares Solution of Nonlinear Problems: Application to Nonlinear Problems in Fluid Dynamics 1 2 3 4 5 19 5 Introduction: Synopsis 19 5 Least-Squares Solution of Finite-Dimensional Systems of Equations 19 5 Least-Squares Solution of a Nonlinear Dirichlet Model Problem 19 8 Transonic Flow Calculations... ||«|| -+ +oo (4 .12 ) 1 Hence (cf Cea [1] , [2]) there exists a unique solution for the optimization problem (n) Characterization of u: We show that the problem (n) is equivalent to (4 .10 ) and thus get a characterization of u (2) Necessity of (4 .10 ): Let 0 < t < 1 Let u be the solution of {%) Then for all v e V we have J(u) < J(u + t(v - u)) (4 .13 ) Set J0(v) = jb(v, v) - L(v), then (4 .13 ) becomes 0 0, V v e U Therefore we have lim inf jh(uh) *->0 < lim inf \a\\uh - u\\2 + jh{uh)~] h-0 < lim sup [a||uh - u|| < a(u, v-u)+ j(v) - L(y - u), V v 6 U (6 .11 ) 7 Penalty Solution of Elliptic Variational Inequalities of the First Kind 15 By the density of U, (6 .11 ) holds, V c e K Replacing v by u in (6 .11 )... (4 .14 ) obtained by using the convexity of j Dividing by t in (4 .14 ) and taking the limit as t -> 0, we get 0 < (J'0(u), i > - « ) + j(v) - j(u), VveV (4 .15 ) Vt,weK (4 .16 ) Since b(-, •) is symmetric, we have (J'0(v), w) = b(v, w) - L(w), From (4 .15 ) and (4 .16 ) we obtain b(u, v - u) + j(v) - j(u) > L(v - u), V v e V This proves the necessity (3) Sufficiency of (4 .10 ): Let u be a solution of (4 .10 ); for . published in 19 84 ISBN 978-3-540-77506-5 DOI 10 .10 07/978-3-540-778 01- 1 e-ISBN 978-3-540-778 01- 1 Scientific Computation ISSN 14 34-8322 Library of Congress Control Number: 2007942575 © 2008, 19 84 Springer-Verlag. Cylindrical Pipe 10 4 CHAPTER IV Applications of Elliptic Variational Inequality Methods to the Solution of Some Nonlinear Elliptic Equations 11 0 1. Introduction 11 0 2. Theoretical and Numerical Analysis. Mildly Nonlinear Elliptic Equations 11 0 3. A Subsonic Flow Problem 13 4 CHAPTER V Relaxation Methods and Applications 14 0 1. Generalities 14 0 2. Some Basic Results of Convex Analysis 14 0 3.