Convergence of a regularized finite element discretization of the two-dimensional Monge-Ampère equation
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866914855027474432 |
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| author | Gallistl, Dietmar Tran, Ngoc Tien |
| author_facet | Gallistl, Dietmar Tran, Ngoc Tien |
| contents | 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 locally 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 locally uniformly convergent. Numerical experiments provide empirical evidence for the efficient approximation of singular solutions $u$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_10711 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Convergence of a regularized finite element discretization of the two-dimensional Monge-Ampère equation Gallistl, Dietmar Tran, Ngoc Tien Numerical Analysis 35J96, 65N12, 65N30, 65Y20 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 locally 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 locally uniformly convergent. Numerical experiments provide empirical evidence for the efficient approximation of singular solutions $u$. |
| title | Convergence of a regularized finite element discretization of the two-dimensional Monge-Ampère equation |
| topic | Numerical Analysis 35J96, 65N12, 65N30, 65Y20 |
| url | https://arxiv.org/abs/2112.10711 |