An alternating direction method of multipliers for the numerical solution of a fully nonlinear partial differential equation involving the Jacobian determinant

Caboussat, Alexandre (Haute école de gestion de Genève, HES-SO // Haute Ecole Spécialisée de Suisse Occidentale) ; Glowinski, Roland (University of Houston, USA)

We consider the Dirichlet problem for a partial differential equation involving the Jacobian determinant in two dimensions of space. The problem consists in finding a vector-valued function such that the determinant of its gradient is given pointwise in a bounded domain, together with essential boundary conditions. This problem was initially considered in Dacorogna and Moser [Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), pp. 1--26], and several theoretical generalizations have been derived since. In this work, we design a numerical algorithm for the approximation of the solution of such a problem for various kinds of boundary data. The proposed method relies on an augmented Lagrangian algorithm with biharmonic regularization, and low order mixed finite element approximations. An iterative method allows us to decouple the nonlinearity and the differential operators. Numerical experiments show the capabilities of the method for benchmarks and then for more demanding test problems.


Keywords:
Article Type:
scientifique
Faculty:
Economie et Services
School:
HEG - Genève
Institute:
CRAG - Centre de Recherche Appliquée en Gestion
Subject(s):
Economie/gestion
Date:
2018
Pagination:
29 p.
Published in
SIAM journal on scientific computing
Numeration (vol. no.):
2018, vol. 40, no. 1, pp. A52–A80
DOI:
ISSN:
1064-8275
Appears in Collection:



 Record created 2018-02-19, last modified 2019-04-11

Fulltext:
Download fulltext
PDF

Rate this document:

Rate this document:
1
2
3
 
(Not yet reviewed)