Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-05-01T18:58:30.929Z Has data issue: false hasContentIssue false
  • English
  • Français

A least-squares method for the numerical solution of the Dirichlet problem for the elliptic monge − ampère equation in dimension two∗

Published online by Cambridge University Press:  03 June 2013

Alexandre Caboussat ,
Roland Glowinski  and
Danny C. Sorensen
Alexandre Caboussat
Affiliation:
Haute École de Gestion/Geneva School of Business Administration, Genève, Switzerland. alexandre.caboussat@hesge.ch University of Houston, Department of Mathematics, 4800 Calhoun Rd, Houston, 77204-3008 Texas, USA; caboussat@math.uh.edu
Roland Glowinski
Affiliation:
University of Houston, Department of Mathematics, 4800 Calhoun Rd, Houston, 77204-3008 Texas, USA; roland@math.uh.edu
Danny C. Sorensen
Affiliation:
Rice University, Department of Computational and Applied Mathematics, MS 134, Houston, 77251-1892 Texas, USA; sorensen@rice.edu
Get access

Abstract

We address in this article the computation of the convex solutions of the Dirichlet problem for the real elliptic Monge − Ampère equation for general convex domains in two dimensions. The method we discuss combines a least-squares formulation with a relaxation method. This approach leads to a sequence of Poisson − Dirichlet problems and another sequence of low dimensional algebraic eigenvalue problems of a new type. Mixed finite element approximations with a smoothing procedure are used for the computer implementation of our least-squares/relaxation methodology. Domains with curved boundaries are easily accommodated. Numerical experiments show the convergence of the computed solutions to their continuous counterparts when such solutions exist. On the other hand, when classical solutions do not exist, our methodology produces solutions in a least-squares sense.

Keywords

Monge − Ampère equationleast-squares methodbiharmonic problemconjugate gradient methodquadratic constraint minimizationmixed finite element methods
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 3 , July 2013 , pp. 780 - 810
Copyright
© EDP Sciences, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aleksandrov, A.D., Uniqueness conditions and estimates for the solution of the Dirichlet problem. Amer. Math. Soc. Trans. 68 (1968) 89–119. Google Scholar
Benamou, J.D., Froese, B.D. and Oberman, A.M., Two numerical methods for the elliptic Monge − Ampère equation. ESAIM: M2AN 44 (2010) 737–758. Google Scholar
M. Bernadou, P.L. George, A. Hassim, P. Joly, P. Laug, A. Perronet, E. Saltel, D. Steer, G. Vanderborck and M. Vidrascu, Modulef, a modular library of finite elements. Technical report, INRIA (1988).
P.B. Bochev and M.D. Gunzburger, Least-Squares Finite Element Methods. Springer-Verlag, New York (2009).
Boehmer, K., On finite element methods for fully nonlinear elliptic equations of second order. SIAM J. Numer. Anal. 46 (2008) 1212–1249. Google Scholar
Brenner, S.C., Gudi, T., Neilan, M. and Sung, L.-Y., c 0 penalty methods for the fully nonlinear Monge − Ampere equation. Math. Comput. 80 (2011) 1979–1995. Google Scholar
Brenner, S.C. and Neilan, M., Finite element approximations of the three dimensional Monge − Ampère equation. ESAIM: M2AN 46 (2012) 979–1001. Google Scholar
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, NewYork (1991).
Caboussat, A. and Glowinski, R., Regularization methods for the numerical solution of the divergence equation ∇·u = f. J. Comput. Math. 30 (2012) 354–380. Google Scholar
Cabré, X., Topics in regularity and qualitative properties of solutions of nonlinear elliptic equations. Discrete Contin. Dyn. Systems 8 (2002) 289–302. Google Scholar
L.A. Caffarelli, Nonlinear elliptic theory and the Monge − Ampère equation, in Proc. of the International Congress of Mathematicians. Higher Education Press, Beijing (2002) 179–187.
L.A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations. American Mathematical Society, Providence, RI (1995).
Caffarelli, L.A. and Glowinski, R., Numerical solution of the Dirichlet problem for a Pucci equation in dimension two. Application to homogenization. J. Numer. Math. 16 (2008) 185–216. Google Scholar
L.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: Application to Geometry and Optimization, American Mathematical Society, Providence, RI (1999) 13–32.
Crandall, M.G., Ishii, H. and Lions, P.-L., User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27 (1992) 1–67. Google Scholar
Dean, E.J. and Glowinski, R., Numerical solution of the two-dimensional elliptic Monge − Ampère equation with Dirichlet boundary conditions: an augmented Lagrangian approach. C. R. Acad. Sci. Paris, Ser. I 336 (2003) 779–784. Google Scholar
Dean, E.J. and Glowinski, R., Numerical solution of the two-dimensional elliptic Monge − Ampère equation with Dirichlet boundary conditions: a least-squares approach. C. R. Acad. Sci. Paris, Ser. I 339 (2004) 887–892. Google Scholar
Dean, E.J. and Glowinski, R., Numerical solution of a two-dimensional elliptic Pucci’s equation with Dirichlet boundary conditions: a least-squares approach. C. R. Acad. Sci. Paris, Ser. I 341 (2005) 374–380. Google Scholar
Dean, E.J. and Glowinski, R., An augmented Lagrangian approach to the numerical solution of the Dirichlet problem for the elliptic Monge − Ampère equation in two dimensions. Electronic Transactions in Numerical Analysis 22 (2006) 71–96. Google Scholar
Dean, E.J. and Glowinski, R., Numerical methods for fully nonlinear elliptic equations of the Monge − Ampère type. Comput. Meth. Appl. Mech. Engrg. 195 (2006) 1344–1386. Google Scholar
E.J. Dean and R. Glowinski, On the numerical solution of the elliptic Monge − Ampère equation in dimension two: A least-squares approach, in Partial Differential Equations: Modeling and Numerical Simulation, vol. 16 of Comput. Methods Appl. Sci., edited by R. Glowinski and P. Neittaanmäki. Springer (2008) 43–63.
E.J. Dean, R. Glowinski and T.W. Pan, Operator-splitting methods and applications to the direct numerical simulation of particulate flow and to the solution of the elliptic Monge − Ampère equation. in Control and Boundary Analysis, edited by J.P. Zolésio J. Cagnol, CRC Boca Raton, FLA (2005) 1–27.
Dean, E.J., Glowinski, R. and Trevas, D., An approximate factorization/least squares solution method for a mixed finite element approximation of the Cahn-Hilliard equation. Jpn J. Ind. Appl. Math. 13 (1996) 495–517. Google Scholar
Feng, X. and Neilan, M., Mixed finite element methods for the fully nonlinear Monge − Ampère equation based on the vanishing moment method. SIAM J. Numer. Anal. 47 (2009) 1226–1250. Google Scholar
Feng, X. and Neilan, M., Vanishing moment method and moment solutions of second order fully nonlinear partial differential equations. J. Sci. Comput. 38 (2009) 74–98. Google Scholar
Froese, B.D. and Oberman, A.M., Convergent finite difference solvers for viscosity solutions of the elliptic Monge − Ampère equation in dimensions two and higher. SIAM J. Numer. Anal. 49 (2011) 1692–1715. Google Scholar
Froese, B.D. and Oberman, A.M., Fast finite difference solvers for singular solutions of the elliptic Monge − Ampère equation. J. Comput. Phys. 230 (2011) 818–834. Google Scholar
Geuzaine, C. and Remacle, J.-F., Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Meth. Eng. 79 (2009) 1309–1331. Google Scholar
D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001).
R. Glowinski, Finite Element Methods For Incompressible Viscous Flow, Handbook of Numerical Analysis, edited by P.G. Ciarlet, J.L. Lions. Elsevier, Amsterdam IX (2003) 3–1176.
R. Glowinski, Numerical Methods for Nonlinear Variational Problems. 2nd edition, Springer-Verlag, New York, NY (2008).
R. Glowinski, Numerical methods for fully nonlinear elliptic equations. in Invited Lectures, 6th Int. Congress on Industrial and Applied Mathematics, Zürich, Switzerland, 16-20 July 2007. EMS (2009) 155–192.
Glowinski, R., Dean, E.J., Guidoboni, G., Juarez, H.L. and Pan, T.W., Applications of operator-splitting methods to the direct numerical simulation of particulate and free surface flows and to the numerical solution of the two-dimensional Monge − Ampère equation. Jpn J. Ind. Appl. Math. 25 (2008) 1–63. Google Scholar
R. Glowinski, J.-L. Lions and J.W. He, Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach. Encyclopedia of Mathematics and its Applications. Cambridge University Press (2008).
Glowinski, R., Marini, D. and Vidrascu, M., Finite-element approximations and iterative solutions of a fourth-order elliptic variational inequality. IMA J. Numer. Anal. 4 (1984) 127–167. Google Scholar
Glowinski, R. and Pironneau, O., Numerical methods for the first bi-harmonic equation and for the two-dimensional Stokes problem. SIAM Rev. 17 (1979) 167–212. Google Scholar
C.E. Gutiérrez, The Monge − Ampère Equation. Birkhaüser, Boston (2001).
Hughes, T.J.R., Franca, L. and Balestra, M., A new finite element formulation for computational fluid dynamics: V. circumventing the Babuska-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolation. Comput. Methods Appl. Mech. Engrg. 59 (1986) 85–100. Google Scholar
Ishii, H. and Lions, P.-L., Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differ. Eq. 83 (1990) 26–78. Google Scholar
Loeper, G. and Rapetti, F., Numerical solution of the Monge − Ampère equation by a Newton’s algorithm. C. R. Math. Acad. Sci. Paris 340 (2005) 319–324. Google Scholar
Mohammadi, B., Optimal transport, shape optimization and global minimization. C. R. Acad Sci Paris, Ser. I 351 (2007) 591–596. Google Scholar
Neilan, M., A nonconforming Morley finite element method for the fully nonlinear Monge − Ampère equation. Numer. Math. 115 (2010) 371–394. Google Scholar
Oberman, A., Wide stencil finite difference schemes for the elliptic Monge − Ampère equations and functions of the eigenvalues of the Hessian. Discr. Contin. Dyn. Syst. B 10 (2008) 221–238. Google Scholar
Oliker, V.I. and Prussner, L.D., On the numerical solution of the equation \hbox{$z_{xx} z_{yy} - z_{xy}^2 = f$}zxxzyy−zxy2=f and its discretization, I. Numer. Math. 54 (1988) 271–293. Google Scholar
Picasso, M., Alauzet, F., Borouchaki, H. and George, P.-L., A numerical study of some Hessian recovery techniques on isotropic and anisotropic meshes. SIAM J. Sci. Comput. 33 (2011) 1058–1076. Google Scholar
A.V. Pogorelov, Monge − Ampère Equations of Elliptic Type. P. Noordhooff, Ltd, Groningen, Netherlands (1964).
Reinhart, L., On the numerical analysis of the Von Kármán equation: mixed finite element approximation and continuation techniques. Numer. Math. 39 (1982) 371–404. Google Scholar
Sorensen, D.C. and Glowinski, R., A quadratically constrained minimization problem arising from PDE of Monge − Ampère type. Numer. Algor. 53 (2010) 53–66. Google Scholar
Tychonoff, A.N., The regularization of incorrectly posed problems. Doklady Akad. Nauk. SSSR 153 (1963) 42–52. Google Scholar
Zheligovsky, V., Podvigina, O. and Frisch, U., The Monge − Ampère equation: Various forms and numerical solution. J. Comput. Phys. 229 (2010) 5043–5061. Google Scholar