TY - JOUR

T1 - An Alternating Direction Method of Multipliers for the Numerical Solution of a Fully Nonlinear Partial Differential Equation Involving the Jacobian Determinant

AU - Caboussat, Alexandre

AU - Glowinski, Roland

N1 - Funding Information:
This work was partially supported by the Swiss National Science Foundation (grant 165785) and the National Science Foundation (grant NSF DMS -0913982).
Publisher copyright:
© 2018, Society for Industrial and Applied Mathematics

PY - 2018/1/5

Y1 - 2018/1/5

N2 - 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.

AB - 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.

KW - ADMM algorithm

KW - Augmented Lagrangian methods

KW - Biharmonic regularization

KW - Finite element method

KW - Jacobian determinant

KW - Quadratically constrained minimization

UR - http://www.scopus.com/inward/record.url?scp=85044535359&partnerID=8YFLogxK

U2 - 10.1137/16M1094075

DO - 10.1137/16M1094075

M3 - Article

AN - SCOPUS:85044535359

SN - 1064-8275

VL - 40

SP - A52-A80

JO - SIAM Journal of Scientific Computing

JF - SIAM Journal of Scientific Computing

IS - 1

ER -