zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Numerical methods for fully nonlinear elliptic equations of the Monge-Ampère type. (English) Zbl 1119.65116
Summary: We discuss the numerical solution of the Dirichlet problem for the Monge-Ampère equation in two dimensions. The solution of closely related problems is also discussed; these include a family of Pucci’s equations, the equation prescribing the harmonic mean of the eigenvalues of the Hessian of a smooth function of two variables, and a minimization problem from nonlinear elasticity, where the cost functional involves the determinant of the gradient of vector-valued functions. To solve the Monge-Ampère equation we consider two methods. The first one reduces the Monge-Ampère equation to a saddle-point problem for a well-chosen augmented Lagrangian; to solve this saddle-point problem we advocate an Uzawa-Douglas-Rachford algorithm. The second method combines nonlinear least-squares and operator-splitting. This second method being simpler to implement, we apply variants of it to the solution of the other problems. For the space discretization we use mixed finite element approximations, closely related to methods already used for the solution of linear and nonlinear bi-harmonic problems; through these approximations the solution of the above problems is, essentially, reduced to the solution of discrete Poisson problems. The methods discussed in this article are validated by the results of numerical experiments.

65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
35J65Nonlinear boundary value problems for linear elliptic equations
UNCMND; pchip
Full Text: DOI
[1] Caffarelli, L. A.: Non linear elliptic theory and the Monge-Ampère equation. Proceedings of the international congress of mathematicians, Beijing 2002, August 20-28, 179-187 (2002) · Zbl 1040.35018
[2] Brenier, Y.: Some geometric pdes related to hydrodynamics and electrodynamics. Proceedings of the international congress of mathematicians, Beijing 2002, August 20-28, 761-771 (2002) · Zbl 1136.37355
[3] Chang, S. -Y.A.; Yang, P. C.: Nonlinear partial differential equations in conformal geometry. Proceedings of the international congress of mathematicians, Beijing 2002, August 20-28, 189-207 (2002) · Zbl 1036.53024
[4] Bao, D.; Chern, S. S.; Shen, Z.: An introduction to Riemann-Finsler geometry. (2000) · Zbl 0954.53001
[5] Caffarelli, L. A.; Cabré, X.: Fully nonlinear elliptic equations. (1995)
[6] Cabré, X.: Topics in regularity and qualitative properties of solutions of nonlinear elliptic equations. Discrete contin. Dynam. syst. 8, No. 2, 289-302 (2002)
[7] Glowinski, R.; Pironneau, O.: Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem. SIAM rev. 17, No. 2, 167-212 (1979) · Zbl 0427.65073
[8] Glowinski, R.; Lions, J. L.; Tremolières, R.: Numerical analysis of variational inequalities. (1981)
[9] Reinhart, L.: On the numerical analysis of the von kármán equations: mixed finite element approximation and continuation techniques. Numer. math. 39, 371-404 (1982) · Zbl 0503.73048
[10] Glowinski, R.; Keller, H. B.; Reinhart, L.: Continuation-conjugate gradient methods for the least-squares solution of nonlinear boundary value problems. SIAM J. Sci. stat. Comp. 4, No. 2, 793-832 (1985) · Zbl 0589.65075
[11] Glowinski, R.; Marini, D.; Vidrascu, M.: Finite element approximation and iterative solution of a fourth-order elliptic variational inequality. IMA J. Numer. anal. 4, 127-167 (1984) · Zbl 0544.65043
[12] Fortin, M.; Glowinski, R.: Augmented Lagrangian methods: application to the numerical solution of boundary value problems. (1983) · Zbl 0525.65045
[13] Glowinski, R.; Le Tallec, P.: Augmented Lagrangians and operator-splitting methods in nonlinear mechanics. (1989) · Zbl 0698.73001
[14] Dean, E. J.; Glowinski, R.; 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, No. 2, 495-517 (1996) · Zbl 0874.65073
[15] Courant, R.; Hilbert, D.: Methods of mathematical physics. (1989) · Zbl 0729.35001
[16] Aubin, T.: Nonlinear analysis on manifolds, Monge-Ampère equations. (1982) · Zbl 0512.53044
[17] Aubin, T.: Some nonlinear problems in riemanian geometry. (1998) · Zbl 0896.53003
[18] Lions, P. L.: Une méthode nouvelle pour l’existence de solutions régulières de l’équation de Monge-Ampère réelle. CR acad. Sci. Paris, sér. I 293, No. 2, 589-592 (1981)
[19] Caffarelli, L. A.; Niremberg, L.; Spruck, J.: The Dirichlet problem for nonlinear second order elliptic equations, (I) Monge-Ampère equation. Commun. pure appl. Math. 37, 369-402 (1984) · Zbl 0598.35047
[20] Caffarelli, L. A.; Kohn, J. J.; Nirenberg, L.; Spruck, J.: The Dirichlet problem for nonlinear second order elliptic equations, (II) complex Monge-Ampère and uniformly elliptic equations. Commun. pure appl. Math. 38, 209-252 (1985) · Zbl 0598.35048
[21] Caffarelli, L. A.; Milman, M.: Monge-Ampère equation: application to geometry and optimization. (1999) · Zbl 0903.00039
[22] Ockendon, J. R.; Howison, S.; Lacey, A.; Movchan, A.: Applied partial differential equations. (1999) · Zbl 0927.35001
[23] Olicker, V. I.; Prussner, L. D.: On the numerical solution of the equation zxxzyy-zxy2=f and its discretization. I. numer. Math. 54, 271-293 (1988)
[24] Caffarelli, L. A.; Kochenkgin, S. A.; Olicker, V. I.: On the numerical solution of reflector design with given far-field scattering data. Monge-Ampère equation: applications to geometry and optimization, 13-32 (1999)
[25] Dean, E. J.; Glowinski, R.: Numerical solution of the two-dimensional elliptic Monge-Ampère equation with Dirichlet boundary conditions: an augmented Lagrangian approach. CR acad. Sci. Paris, ser. I 336, 779-784 (2003) · Zbl 1028.65120
[26] Dean, E. J.; Glowinski, R.; Pan, T. W.: Operator-splitting methods and applications to the direct numerical simulation of particulate flow and to the solution of the elliptic Monge-Ampère equation. Control and boundary analysis, 1-27 (2005) · Zbl 1188.65115
[27] Glowinski, R.: Numerical methods for nonlinear variational problems. (1984) · Zbl 0536.65054
[28] Glowinski, R.: Finite element methods for incompressible viscous flow. Handbook of numerical analysis, 3-1176 (2003) · Zbl 1040.76001
[29] Dennis, J. E.; Schnabel, R. B.: Numerical methods for unconstrained optimization and nonlinear equations. (1996) · Zbl 0847.65038
[30] Evans, L. C.: Partial differential equations and Monge-Kantorovich mass transfer. Current developments in mathematics, 26-78 (1997)
[31] Benamou, J. D.; Brenier, Y.: A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. math. 84, 375-393 (2000) · Zbl 0968.76069
[32] Ciarlet, P. G.: The finite element method for elliptic problems. (1978) · Zbl 0383.65058
[33] Brezzi, F.; Fortin, M.: Mixed and hybrid finite element methods. (1991) · Zbl 0788.73002
[34] Ciarlet, P. G.: Basic error estimates. Handbook of numerical analysis, 17-351 (1991) · Zbl 0875.65086
[35] Marchuk, G. I.: Splitting and alternating directions methods. Handbook of numerical analysis, 197-462 (1990) · Zbl 0875.65049
[36] Lions, P. L.; Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. anal. 16, No. 2, 964-979 (1979) · Zbl 0426.65050
[37] Dieudonné, J.: Panorama des mathématiques pures: le choix bourbachique. (2003) · Zbl 0482.00002
[38] Kahaner, D.; Moler, C.; Nash, S.: Numerical methods of software. (1988) · Zbl 0744.65002
[39] Schnabel, R. B.; Koontz, J. E.; Weiss, B. E.: A modular system of algorithms for unconstrained minimization. ACM trans. Math. softw. 11, No. 2, 419-440 (1985) · Zbl 0591.65045
[40] Marsden, J. E.; Hughes, T. J. R.: Mathematical foundations of elasticity. (1994) · Zbl 0545.73031
[41] Chorin, A. J.; Hughes, T. J. R.; Mccracken, M. F.; Marsden, J. E.: Product formulas and numerical algorithms. Commun. pure appl. Math. 31, 205-256 (1978) · Zbl 0358.65082