Issue |
ESAIM: COCV
Volume 26, 2020
Special issue in honor of Enrique Zuazua's 60th birthday
|
|
---|---|---|
Article Number | 118 | |
Number of page(s) | 25 | |
DOI | https://doi.org/10.1051/cocv/2020072 | |
Published online | 17 December 2020 |
On the numerical solution of nonlinear eigenvalue problems for the Monge-Ampère operator*
1
Department of Mathematics, University of Houston,
4800 Calhoun Road,
Houston,
TX
77204, USA.
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: hao.liu@math.gatech.edu
Received:
20
August
2020
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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