Geodesic
On the numerical solution of nonlinear eigenvalue problems for the Monge-Ampère operator
ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 118.

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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.

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DOI : 10.1051/cocv/2020072
Classification : 35J60, 65N25, 65N30
Keywords: Monge-Ampère equation, nonlinear eigenvalue problems, operator-splitting methods, finite element approximations 
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     author = {Glowinski, Roland and Leung, Shingyu and Liu, Hao and Qian, Jianliang},
     editor = {Buttazzo, G. and Casas, E. and de Teresa, L. and Glowinsk, R. and Leugering, G. and Tr\'elat, E. and Zhang, X.},
     title = {On the numerical solution of nonlinear eigenvalue problems for the {Monge-Amp\`ere} operator},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {26},
     year = {2020},
     doi = {10.1051/cocv/2020072},
     mrnumber = {4188832},
     zbl = {1460.35183},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2020072/}
}
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Glowinski, Roland; Leung, Shingyu; Liu, Hao; Qian, Jianliang. On the numerical solution of nonlinear eigenvalue problems for the Monge-Ampère operator. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 118. doi : 10.1051/cocv/2020072. http://geodesic.mathdoc.fr/articles/10.1051/cocv/2020072/

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Dedicated to Enrique Zuazua on the occasion of his 60th birthday.