Stability and guaranteed error control of approximations to the Monge--Ampère equation
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866913412363059200 |
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| author | Gallistl, Dietmar Tran, Ngoc Tien |
| author_facet | Gallistl, Dietmar Tran, Ngoc Tien |
| contents | This paper analyzes a regularization scheme of the Monge--Ampère equation by uniformly elliptic Hamilton--Jacobi--Bellman equations. The main tools are stability estimates in the $L^\infty$ norm from the theory of viscosity solutions which are independent of the regularization parameter $\varepsilon$. They allow for the uniform convergence of the solution $u_\varepsilon$ to the regularized problem towards the Alexandrov solution $u$ to the Monge--Ampère equation for any nonnegative $L^n$ right-hand side and continuous Dirichlet data. The main application are guaranteed a posteriori error bounds in the $L^\infty$ norm for continuously differentiable finite element approximations of $u$ or $u_\varepsilon$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_06805 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Stability and guaranteed error control of approximations to the Monge--Ampère equation Gallistl, Dietmar Tran, Ngoc Tien Numerical Analysis This paper analyzes a regularization scheme of the Monge--Ampère equation by uniformly elliptic Hamilton--Jacobi--Bellman equations. The main tools are stability estimates in the $L^\infty$ norm from the theory of viscosity solutions which are independent of the regularization parameter $\varepsilon$. They allow for the uniform convergence of the solution $u_\varepsilon$ to the regularized problem towards the Alexandrov solution $u$ to the Monge--Ampère equation for any nonnegative $L^n$ right-hand side and continuous Dirichlet data. The main application are guaranteed a posteriori error bounds in the $L^\infty$ norm for continuously differentiable finite element approximations of $u$ or $u_\varepsilon$. |
| title | Stability and guaranteed error control of approximations to the Monge--Ampère equation |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2301.06805 |