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arXiv · 2008.08103

On the Numerical Solution of Nonlinear Eigenvalue Problems for the Monge-Amp\`{e}re Operator

Roland Glowinski·Shingyu Leung·Hao Liu·Jianliang Qian
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Abstract

In this article, we report the results we obtained when investigating the numerical solution of some nonlinear eigenvalue problems for the Monge-Amp\`{e}re operator $v\rightarrow \det \mathbf{D}^2 v$. 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 $h\rightarrow 0$. We considered also test problems with no known exact solutions.

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Roland Glowinski, Shingyu Leung, Hao Liu, Jianliang Qian. 2020-08-18. On the Numerical Solution of Nonlinear Eigenvalue Problems for the Monge-Amp\`{e}re Operator. https://arxiv.org/abs/2008.08103

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