Volume 26, 2020
Special issue in the honor of Enrique Zuazua's 60th birthday
|Number of page(s)||25|
|Published online||17 December 2020|
On the numerical solution of nonlinear eigenvalue problems for the Monge-Ampère operator*
Department of Mathematics, University of Houston,
4800 Calhoun Road,
2 Department of Mathematics, the Hong Kong Baptist University, Hong Kong.
3 Department of Mathematics, the Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong.
4 School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332, USA.
5 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA.
** Corresponding author: email@example.com
Accepted: 19 October 2020
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.
Mathematics Subject Classification: 35J60 / 65N25 / 65N30
Key words: Monge-Ampère equation / nonlinear eigenvalue problems / operator-splitting methods / finite element approximations
© EDP Sciences, SMAI 2020
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