zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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.

MSC:
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
35J65Nonlinear boundary value problems for linear elliptic equations