TY - JOUR
T1 - On the numerical solution of nonlinear eigenvalue problems for the Monge-Ampère operator
AU - Glowinski, Roland
AU - Leung, Shingyu
AU - Liu, Hao
AU - Qian, Jianliang
N1 - Funding Information:
Acknowledgements. R. Glowinski acknowledges the support of the Hong Kong based Kennedy Wong Foundation. J. Qian is partially supported by NSF grants. S. Leung is supported by the Hong Kong RGC under Grant 16302819.
PY - 2020/12/17
Y1 - 2020/12/17
N2 - In this article, we report the results we obtained when investigating the numerical solution of some nonlinear eigenvalue problems for the Monge-Ampère operator v → det D2v. The methodology we employ relies on the following ingredients: (i) a divergence formulation of the eigenvalue problems under consideration. (ii) The time discretization by operator-splitting of an initial value problem (a kind of gradient flow) associated with each eigenvalue problem. (iii) A finite element approximation relying on spaces of continuous piecewise affine functions. To validate the above methodology, we applied it to the solution of problems with known exact solutions: The results we obtained suggest convergence to the exact solution when the space discretization step h → 0. We considered also test problems with no known exact solutions.
AB - In this article, we report the results we obtained when investigating the numerical solution of some nonlinear eigenvalue problems for the Monge-Ampère operator v → det D2v. The methodology we employ relies on the following ingredients: (i) a divergence formulation of the eigenvalue problems under consideration. (ii) The time discretization by operator-splitting of an initial value problem (a kind of gradient flow) associated with each eigenvalue problem. (iii) A finite element approximation relying on spaces of continuous piecewise affine functions. To validate the above methodology, we applied it to the solution of problems with known exact solutions: The results we obtained suggest convergence to the exact solution when the space discretization step h → 0. We considered also test problems with no known exact solutions.
KW - Finite element approximations
KW - Monge-Ampère equation
KW - Nonlinear eigenvalue problems
KW - Operator-splitting methods
UR - http://www.scopus.com/inward/record.url?scp=85098159336&partnerID=8YFLogxK
U2 - 10.1051/cocv/2020072
DO - 10.1051/cocv/2020072
M3 - Journal article
AN - SCOPUS:85098159336
SN - 1292-8119
VL - 26
JO - ESAIM - Control, Optimisation and Calculus of Variations
JF - ESAIM - Control, Optimisation and Calculus of Variations
M1 - 118
ER -