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 - Article

AN - SCOPUS:85098159336

VL - 26

JO - ESAIM - Control, Optimisation and Calculus of Variations

JF - ESAIM - Control, Optimisation and Calculus of Variations

SN - 1292-8119

M1 - 118

ER -