On the Numerical Solution of Nonlinear Eigenvalue Problems for the Monge-Ampère Operator

2020/08/18 by Roland Glowinski, Glowinski, Roland, Shingyu Leung +5 · 1 citation
Mathematics · #35J60 #65N25 #65N30 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2008.08103

openalex publication_date 2020/08/18 · openalex created_date 2020/08/24 · openalex updated_date 2026/07/28

Abstract

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 D² 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→ 0. We considered also test problems with no known exact solutions.

Citations

• The eigenvalue problem for the Monge-Amp `ere operator on general\n bounded convex domains

Cited by

• Fast Operator-Splitting Methods for Nonlinear Elliptic Equations

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