[Une méthode de lagrangien augmenté pour la résolution numérique du problème de Monge–Ampère elliptique en dimension deux avec conditions de Dirichlet]
The main goal of this Note is to discuss a method for the numerical solution of the two-dimensional elliptic Monge–Ampère equation with Dirichlet boundary conditions (the E-MAD problem). This method relies on the reformulation of E-MAD as a problem of Calculus of Variation involving the biharmonic operator (or closely related operators), and then to a saddle-point formulation for a well-chosen augmented Lagrangian functional, leading to iterative methods such as Uzawa–Douglas–Rachford. The above methodology applies to problems other than E-MAD (such as the Pucci equation). The results of numerical experiments are presented. They concern the solution of E-MAD on the unit square (0,1)×(0,1); the first test problem has a known smooth closed form solution which is easily computed with optimal order of convergence. The second test problem has also a known closed form solution; the fact that this solution has the
L'objet essentiel de cette Note est l'étude d'une méthode pour la résolution numérique du problème de Dirichlet pour l'équation de Monge–Ampère elliptique en dimension deux (le problème E-MAD). Cette méthode repose sur une reformulation de E-MAD comme un problème de Calcul des Variations impliquant l'opérateur bi-harmonique (ou des opérateurs voisins), puis sur une formulation de type point-selle pour un Lagrangien augmenté bien choisi, ce qui conduit naturellement à des algorithmes du type Uzawa–Douglas–Rachford. La méthodologie ci-dessus s'applique à des problèmes autres que E-MAD (l'équation de Pucci, par exemple). Les résultats d'essais numériques sont egalement presentés. Ils concernent la résolution du problème E-MAD sur le carré unité (0,1)×(0,1). Le premier problème test a une solution régulière (analytique, en fait) connue exactement ; on la retrouve facilement, avec une erreur d'approximation d'ordre optimal. La solution du second probleme test est aussi connue exactement ; le fait qu'elle soit dans
Accepté le :
Publié le :
Edward J. Dean 1 ; Roland Glowinski 1
@article{CRMATH_2003__336_9_779_0, author = {Edward J. Dean and Roland Glowinski}, title = {Numerical solution of the two-dimensional elliptic {Monge{\textendash}Amp\`ere} equation with {Dirichlet} boundary conditions: an augmented {Lagrangian} approach}, journal = {Comptes Rendus. Math\'ematique}, pages = {779--784}, publisher = {Elsevier}, volume = {336}, number = {9}, year = {2003}, doi = {10.1016/S1631-073X(03)00149-3}, language = {en}, }
TY - JOUR AU - Edward J. Dean AU - Roland Glowinski TI - Numerical solution of the two-dimensional elliptic Monge–Ampère equation with Dirichlet boundary conditions: an augmented Lagrangian approach JO - Comptes Rendus. Mathématique PY - 2003 SP - 779 EP - 784 VL - 336 IS - 9 PB - Elsevier DO - 10.1016/S1631-073X(03)00149-3 LA - en ID - CRMATH_2003__336_9_779_0 ER -
%0 Journal Article %A Edward J. Dean %A Roland Glowinski %T Numerical solution of the two-dimensional elliptic Monge–Ampère equation with Dirichlet boundary conditions: an augmented Lagrangian approach %J Comptes Rendus. Mathématique %D 2003 %P 779-784 %V 336 %N 9 %I Elsevier %R 10.1016/S1631-073X(03)00149-3 %G en %F CRMATH_2003__336_9_779_0
Edward J. Dean; Roland Glowinski. Numerical solution of the two-dimensional elliptic Monge–Ampère equation with Dirichlet boundary conditions: an augmented Lagrangian approach. Comptes Rendus. Mathématique, Volume 336 (2003) no. 9, pp. 779-784. doi : 10.1016/S1631-073X(03)00149-3. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00149-3/
[1] Nonlinear Analysis on Manifolds, Springer-Verlag, Berlin, 1982
[2] L.A. Caffarelli, The Monge–Ampère equation and optimal transportation: an elementary review, Lecture at ICM 2002, Beijing, August 20–28, 2002
[3] Fully Nonlinear Elliptic Equations, American Mathematical Society, Providence, RI, 1995
[4] Augmented Lagrangians, North-Holland, Amsterdam, 1983
[5] Augmented Lagrangians and Operator Splitting Methods in Nonlinear Mechanics, SIAM, Philadelphia, 1989
[6] Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981
[7] On the numerical solution of the equation zxxzyy−zxy2=f and its discretization, I, Numer. Math., Volume 54 (1988), pp. 271-293
- A nonlinear least-squares convexity enforcing 𝐶⁰ interior penalty method for the Monge–Ampère equation on strictly convex smooth planar domains, Communications of the American Mathematical Society, Volume 4 (2024) no. 14, p. 607 | DOI:10.1090/cams/39
- Adaptive Isogeometric Analysis using optimal transport and their fast solvers, Computer Methods in Applied Mechanics and Engineering, Volume 418 (2024), p. 116570 | DOI:10.1016/j.cma.2023.116570
- Operator-Splitting/Finite Element Methods for the Minkowski Problem, SIAM Journal on Scientific Computing, Volume 46 (2024) no. 5, p. A3230 | DOI:10.1137/23m1590779
- Designing funicular grids with planar quads using isotropic Linear-Weingarten surfaces, International Journal of Solids and Structures, Volume 264 (2023), p. 112028 | DOI:10.1016/j.ijsolstr.2022.112028
- Trivariate Spline Collocation Methods for Numerical Solution to 3D Monge-Ampère Equation, Journal of Scientific Computing, Volume 95 (2023) no. 2 | DOI:10.1007/s10915-023-02183-9
- A C0 linear finite element method for a second‐order elliptic equation in non‐divergence form with Cordes coefficients, Numerical Methods for Partial Differential Equations, Volume 39 (2023) no. 3, p. 2244 | DOI:10.1002/num.22965
- A convexity enforcing
interior penalty method for the Monge–Ampère equation on convex polygonal domains, Numerische Mathematik, Volume 148 (2021) no. 3, p. 497 | DOI:10.1007/s00211-021-01210-x - Unified mathematical framework for a class of fundamental freeform optical systems, Optics Express, Volume 29 (2021) no. 20, p. 31650 | DOI:10.1364/oe.438920
- The Monge–Ampère equation, Geometric Partial Differential Equations - Part I, Volume 21 (2020), p. 105 | DOI:10.1016/bs.hna.2019.05.003
- A note on the Monge–Ampère type equations with general source terms, Mathematics of Computation, Volume 89 (2020) no. 326, p. 2675 | DOI:10.1090/mcom/3554
- A Newton Div-Curl Least-Squares Finite Element Method for the Elliptic Monge–Ampère Equation, Computational Methods in Applied Mathematics, Volume 19 (2019) no. 3, p. 631 | DOI:10.1515/cmam-2018-0196
- A Finite Element/Operator-Splitting Method for the Numerical Solution of the Two Dimensional Elliptic Monge–Ampère Equation, Journal of Scientific Computing, Volume 79 (2019) no. 1, p. 1 | DOI:10.1007/s10915-018-0839-y
- Pointwise rates of convergence for the Oliker–Prussner method for the Monge–Ampère equation, Numerische Mathematik, Volume 141 (2019) no. 1, p. 253 | DOI:10.1007/s00211-018-0988-9
- Convergence Rate Estimates for Aleksandrov's Solution to the Monge–Ampère Equation, SIAM Journal on Numerical Analysis, Volume 57 (2019) no. 1, p. 173 | DOI:10.1137/18m1197217
- Optimization approach for the Monge-Ampère equation, Acta Mathematica Scientia, Volume 38 (2018) no. 4, p. 1285 | DOI:10.1016/s0252-9602(18)30814-2
- Discrete ABP Estimate and Convergence Rates for Linear Elliptic Equations in Non-divergence Form, Foundations of Computational Mathematics, Volume 18 (2018) no. 3, p. 537 | DOI:10.1007/s10208-017-9347-y
- An Accelerated Method for Nonlinear Elliptic PDE, Journal of Scientific Computing, Volume 69 (2016) no. 2, p. 556 | DOI:10.1007/s10915-016-0215-8
- ADMM and Non-convex Variational Problems, Splitting Methods in Communication, Imaging, Science, and Engineering (2016), p. 251 | DOI:10.1007/978-3-319-41589-5_8
- Spline element method for Monge–Ampère equations, BIT Numerical Mathematics, Volume 55 (2015) no. 3, p. 625 | DOI:10.1007/s10543-014-0524-y
- Solving the Monge–Ampère equations for the inverse reflector problem, Mathematical Models and Methods in Applied Sciences, Volume 25 (2015) no. 05, p. 803 | DOI:10.1142/s0218202515500190
- Convergent finite difference methods for one-dimensional fully nonlinear second order partial differential equations, Journal of Computational and Applied Mathematics, Volume 254 (2013), p. 81 | DOI:10.1016/j.cam.2013.02.001
- Quadratic Finite Element Approximations of the Monge-Ampère Equation, Journal of Scientific Computing, Volume 54 (2013) no. 1, p. 200 | DOI:10.1007/s10915-012-9617-4
- A Finite Element Method for Nonlinear Elliptic Problems, SIAM Journal on Scientific Computing, Volume 35 (2013) no. 4, p. A2025 | DOI:10.1137/120887655
- Recent Developments in Numerical Methods for Fully Nonlinear Second Order Partial Differential Equations, SIAM Review, Volume 55 (2013) no. 2, p. 205 | DOI:10.1137/110825960
- An efficient approach for the numerical solution of the Monge–Ampère equation, Applied Numerical Mathematics, Volume 61 (2011) no. 3, p. 298 | DOI:10.1016/j.apnum.2010.10.006
- On the Numerical Simulation of Viscoplastic Fluid Flow, Numerical Methods for Non-Newtonian Fluids, Volume 16 (2011), p. 483 | DOI:10.1016/b978-0-444-53047-9.00006-x
- Looking for the best constant in a Sobolev inequality: a numerical approach, Calcolo, Volume 47 (2010) no. 4, p. 211 | DOI:10.1007/s10092-010-0020-y
- The Monge–Ampère equation: Various forms and numerical solution, Journal of Computational Physics, Volume 229 (2010) no. 13, p. 5043 | DOI:10.1016/j.jcp.2010.03.025
- General solution of two-dimensional beam-shaping with two surfaces, Journal of Physics: Conference Series, Volume 206 (2010), p. 012021 | DOI:10.1088/1742-6596/206/1/012021
- Numerical Methods for the Vector-Valued Solutions of Non-smooth Eigenvalue Problems, Journal of Scientific Computing, Volume 45 (2010) no. 1-3, p. 64 | DOI:10.1007/s10915-010-9383-0
- Adaptivity with moving grids, Acta Numerica, Volume 18 (2009), p. 111 | DOI:10.1017/s0962492906400015
- Vanishing Moment Method and Moment Solutions for Fully Nonlinear Second Order Partial Differential Equations, Journal of Scientific Computing, Volume 38 (2009) no. 1, p. 74 | DOI:10.1007/s10915-008-9221-9
- 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 elliptic Monge-Ampère equation, Japan Journal of Industrial and Applied Mathematics, Volume 25 (2008) no. 1, p. 1 | DOI:10.1007/bf03167512
- On the Numerical Solution of the Elliptic Monge—Ampère Equation in Dimension Two: A Least-Squares Approach, Partial Differential Equations, Volume 16 (2008), p. 43 | DOI:10.1007/978-1-4020-8758-5_3
- On Finite Element Methods for Fully Nonlinear Elliptic Equations of Second Order, SIAM Journal on Numerical Analysis, Volume 46 (2008) no. 3, p. 1212 | DOI:10.1137/040621740
- Sur la permanence de l’œuvre de Gaspard Monge dans la science des 20ème et 21ème siècles, Bulletin de la Sabix (2007) no. 41, p. 81 | DOI:10.4000/sabix.112
- Numerical methods for fully nonlinear elliptic equations of the Monge–Ampère type, Computer Methods in Applied Mechanics and Engineering, Volume 195 (2006) no. 13-16, p. 1344 | DOI:10.1016/j.cma.2005.05.023
- , AIAA Space 2003 Conference Exposition (2003) | DOI:10.2514/6.2003-6322
Cité par 38 documents. Sources : Crossref
Commentaires - Politique