Paper Info
Title
An Alternating Direction Method of Multipliers for the Numerical Solution of a Fully Nonlinear Partial Differential Equation Involving the Jacobian Determinant.
Abstract
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. Poincare Anal. Non Lineaire, 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.
Year
DOI
Venue
2018
10.1137/16M1094075
SIAM JOURNAL ON SCIENTIFIC COMPUTING
Keywords
Field
DocType
Jacobian determinant,augmented Lagrangian methods,ADMM algorithm,biharmonic regularization,finite element method,quadratically constrained minimization
Boundary value problem,Jacobian matrix and determinant,Dirichlet problem,Mathematical analysis,Iterative method,Finite element method,Augmented Lagrangian method,Biharmonic equation,Partial differential equation,Mathematics
Journal
Volume
Issue
ISSN
40
1
1064-8275
Citations 
PageRank 
References 
1
0.37
10
Authors
2
Name
Order
Citations
PageRank
Alexandre Caboussat1226.24
Roland Glowinski218850.44