An augmented Lagrangian approach to the numerical solution of the dirichlet problem for the elliptic Monge-Ampère equation in two dimensions

Electronic Transactions on Numerical Analysis

Volumn 22, Issue , 2006, Pages 71-96

  Dean, E J ^a   Glowinski, R ^a  

^a University of Houston   (United States)

Author keywords

Augmented Lagrangian algorithms; Elliptic Monge Amp re equation; Mixed finite element approximations

Indexed keywords

ALGORITHMS; APPROXIMATION THEORY; CONVERGENCE OF NUMERICAL METHODS; DIFFERENTIAL EQUATIONS; FINITE ELEMENT METHOD; ITERATIVE METHODS; NUMERICAL ANALYSIS; PROBLEM SOLVING;

AUGMENTED LAGRANGIAN ALGORITHMS; ELLIPTIC MONGE-AMPÈRE EQUATION; MIXED FINITE ELEMENT APPROXIMATIONS;

LAGRANGE MULTIPLIERS;

EID: 33646378906     PISSN: 10689613     EISSN: 10689613     Source Type: Journal    
DOI: None     Document Type: Article

References (42)

1. L. A. CAFFARELLI, Nonlinear elliptic theory and the Monge-Ampère equation, in Proceedings of the International Congress of Mathematicians, Beijing 2002, August 20-28, Vol. I, Higher Education Press, Beijing, 2002, pp. 179-187.
2. Y. BRENIER, Some geometric PDEs related to Hydrodynamics and Electrodynamics, in Proceedings of the International Congress of Mathematicians, Beijing 2002, August 20-28, Vol. III, Higher Education Press, Beijing, 2002, pp. 761-771.
3. S-Y. A. CHANG AND P. C. YANG, Nonlinear partial differential equations in conformal geometry, in Proceedings of the International Congress of Mathematicians, Beijing 2002, August 20-28, Vol. I, Higher Education Press, Beijing, 2002, pp. 189-207.
4. D. BAO, S.-S. CHERN, AND Z. SHEN, An Introduction to Riemann-Finsler Geometry,Graduate Texts in Mathematics, 200. Springer-Verlag, New York, 2000.
5. R. GLOWINSKI AND O. PIRONNEAU, Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem, SIAM Rev., 17 (1979), pp. 167-212.
6. R. GLOWINSKI, J. L. LIONS, AND R. TREMOLIÉRES, Numerical Analysis of Variational Inequalities, Translated from the French. Studies in Mathematics and its Applications, 8. North-Holland Publishing Co., Amsterdam-New York, 1981.
7. L. REINHART, On the numerical analysis of the Von Kármán equations: Mixed finite element approximation and continuation techniques, Numer, Math., 39 (1982), pp. 371-404.
8. R. GLOWINSKI, H. B. KELLER, AND L. REINHART, Continuation-conjugate gradient methods for the least-squares solution of nonlinear boundary value problems, SIAM J. Sci. Stat. Comp., 4 (1985), pp. 793-832.
9. R. GLOWINSKI, D. MARINI, AND M. VIDRASCU, Finite element approximation and iterative solution of a fourth-order elliptic variational inequality, IMA J. Numer. Anal., 4 (1984), pp. 127-167.
10. M. FORTIN AND R. GLOWINSKI, Augmented Lagrangian Methods: Application to the Numerical Solution of Boundary Value Problems, North-Holland, Amsterdam, 1983.
11. R. GLOWINSKI AND P. LE TALLEC, Augmented Lagrangians and Operator-Splitting Methods in Nonlinear Mechanics, SIAM, Philadelphia, PA, 1989.
12. E. J. DEAN, R. GLOWINSKI, AND D. TREVAS, An approximate factorization / least-squares solution method for a mixed finite element approximation of the Cahn- Hilliard equation, Japan J. Industr. Appl. Math., 13 (1996), pp. 495-517.
13. L. A. CAFFARELLI AND X. CABRÉ, Fully Nonlinear Elliptic Equations, American Mathematical Society, Providence, RI, 1995.
14. X. CABRÉ, Topics in regularity and qualitative properties of solutions of nonlinear elliptic equations,Current developments in partial differential equations (Temuco, 1999). Discrete Contin. Dyn. Syst. 8 (2002), pp. 289-302.
15. R. COURANT AND D. HILBERT, Methods of Mathematical Physics, Vol. II, Wiley Interscience, New York, NY, 1989.
16. T. AUBIN, Nonlinear Analysis on Manifolds, Monge-Ampère Equations. Springer-Verlag, Berlin, 1982.
17. T. AUBIN, Some Nonlinear Problems in Riemannian Geometry, Springer-Verlag, 1998.
18. P. L. LIONS, Une méthode nouvelle pour l'existence de solutions réguliéres de l'équation de Monge-Ampère réelle, C. R. Acad. Sci. Paris, Série I, 293 (1981), pp. 589-592.
19. L. A. CAFFARELLI, L. NIRENBERG, AND J. SPRUCK, The Dirichlet problem for nonlinear second order elliptic equations, (I) Monge-Ampère equation, Comm. Pure Appl. Math., 37 (1984), pp. 369-402.
20. L. A. CAFFARELLI, J. J. KOHN, L. NIRENBERG, AND J. SPRUCK, The Dirichlet problem for nonlinear second order elliptic equations, (II) Complex Monge-Ampère and uniformly elliptic equations, Comm. Pure Appl. Math., 38 (1985), pp. 209-252.
21. L. A. CAFFARELLI AND Y. Y. LI, A Liouville theorem for solutions of the Monge-Ampère equation with periodic data. Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), pp. 97-120.
22. F. X. LE DIMET AND M. OUBERDOUS, Retrieval of balanced fields: an optimal control method, Tellus, 45A (1993), pp.449-461.
23. L. A. CAFFARELLI AND M. MILMAN (EDS.), Monge-Ampère Equation: Application to Geometry and Optimization, American Math. Society, Providence, RI, 1999.
24. J. R. OCKENDON, S. HOWISON, A. LACEY, AND A. MOVCHAN (EDS.), Applied Partial Differential Equations, Oxford University Press, 1999, Oxford, UK.
25. U. A. CAFFARELLI, S. A. KOCHENKGIN, AND V. I. OLICKER, On the numerical solution of reflector design with given far-field scattering data, in Monge Ampère Equation: Applications to Geometry and Optimization, L. A. Caffarelli and M. Milman, eds., American Mathematical Society, Providence. RI, 1999, pp. 13-32.
26. 2 = f and its discretization, Numer. Math., 54 (1988), pp. 271-293.
27. E. J. DEAN AND R. GLOWINSKI, Numerical solution of the two-dimensional elliptic Monge-Ampère equation with Dirichlet boundary conditions: an augmented Ltigrangian approach, C. R. Acad. Sci. Paris, Serie I, 336 (2003), pp. 779-784.
28. S. V. KOCHENGIN AND V. I. OLIKER, Determination of reflector surfaces from near-field scattering data, Numer. Math., 79 (1998), pp. 553-568.
29. E. J. DEAN, R. GLOWINSKI, AND T. W. PAN, Operator-splitting methods and application to the direct numerical simulation of particulate flow and to the solution of the elliptic Monge-Ampère equation, in Control and Boundary Analysis, J. Cagnol and J. P. Zolésio, eds, Marcel Dekker, New York, NY, 2005.
30. R. GLOWINSKI, Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York, NY, 1984.
31. R. GLOWINSKI, Finite element methods for incompressible viscous flow, in Handbook of Numerical Analysis, Volume IX. P.G. Ciarlet and J. L. Lions, eds., North-Holland, Amsterdam, 2003, pp. 3-1176.
32. J. E. DENNIS, AND R. B. SCHNABEL, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, SIAM, Philadelphia, PA, 1996.
33. L. C. EVANS, Partial differential equations and Monge-Kantorovich mass transfer, in Current Developments in Mathematics, R. Bolt, ed., International Press, Cambridge, MA, 1997, pp.26-78.
34. J. D. BENANOU AND Y. BRENIER, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), pp. 375-393.
35. P. G. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
36. F. BREZZI AND M. FORTIN, Mixed and Hybrid Finite Element Methods, Springer- Verlag, New-York, NY, 1991.
37. P. G. CIARLET, Basic error estimates, in Handbook of Numerical Analysis, Volume II. P. G. Ciarlet and J. L. Lions eds., North-Holland, Amsterdam, 1991, pp. 17-351.
38. E. J. DEAN AND R. GLOWINSKI, Numerical solution of the two-dimensional elliptic Monge-Ampère equation with Dirichlet boundary conditions: a least squares approach. C. R. Acad. Sci. Paris, Série I, 339 (2004), pp.887-892.
39. J. DIEUDONNÉ, Panorama des Mathématiques Pures: Le Choix Bourbachique, Editions Jacques Gabay, Paris, 2003.
40. R. JENSEN, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch. Rat. Mech. Anal., 101 (1988), pp. 1-27.
41. J. I. E. URBAS, Regularity of generalized solutions of Monge-Ampère equations, Math. Z., 197 (1988), pp. 365-393.
42. M. G. CRANDALL, H. ISHII, AND P. L. LIONS, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), pp. 1-67.