## Convergence of a regularized finite element discretization of the two-dimensional Monge–Ampère equation

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Dietmar Gallistl and Ngoc Tien Tran
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## Abstract:

This paper proposes a regularization of the Monge–Ampère equation in planar convex domains through uniformly elliptic Hamilton–Jacobi–Bellman equations. The regularized problem possesses a unique strong solution $u_\varepsilon$ and is accessible to the discretization with finite elements. This work establishes uniform convergence of $u_\varepsilon$ to the convex Alexandrov solution $u$ to the Monge–Ampère equation as the regularization parameter $\varepsilon$ approaches $0$. A mixed finite element method for the approximation of $u_\varepsilon$ is proposed, and the regularized finite element scheme is shown to be uniformly convergent. The class of admissible right-hand sides are the functions that can be approximated from below by positive continuous functions in the $L^1$ norm. Numerical experiments provide empirical evidence for the efficient approximation of singular solutions $u$.## References

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## Additional Information

**Dietmar Gallistl**- Affiliation: Institut für Mathematik, Friedrich-Schiller-Universität Jena, 07743 Jena, Germany
- MR Author ID: 1020312
- Email: dietmar.gallistl@uni-jena.de
**Ngoc Tien Tran**- Affiliation: Institut für Mathematik, Friedrich-Schiller-Universität Jena, 07743 Jena, Germany
- MR Author ID: 1437571
- Email: ngoc.tien.tran@uni-jena.de
- Received by editor(s): December 20, 2021
- Received by editor(s) in revised form: June 7, 2022, July 26, 2022, and September 11, 2022
- Published electronically: February 6, 2023
- Additional Notes: This project received funding from the European Union’s Horizon 2020 research and innovation programme (project DAFNE, grant agreement No. 891734).
- © Copyright 2023 American Mathematical Society
- Journal: Math. Comp.
**92**(2023), 1467-1490 - MSC (2020): Primary 35J96, 65N12, 65N30, 65Y20
- DOI: https://doi.org/10.1090/mcom/3794
- MathSciNet review: 4570330