Error estimates and adaptivity for a least-squares method applied to the Monge-Ampère equation
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916936725561344 |
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| author | Caboussat, Alexandre Peruso, Anna Picasso, Marco |
| author_facet | Caboussat, Alexandre Peruso, Anna Picasso, Marco |
| contents | We introduce novel a posteriori error indicators for a nonlinear least-squares solver for smooth solutions of the Monge--Ampère equation on convex polygonal domains in $\mathbb{R}^2$. At each iteration, our iterative scheme decouples the problem into (i) a pointwise nonlinear minimization problem and (ii) a linear biharmonic variational problem. For the latter, we derive an equivalence to a biharmonic problem with Navier boundary conditions and solve it via mixed piecewise-linear finite elements. Reformulating this as a coupled second-order system, we derive a priori and a posteriori $\mathbb{P}^1$ finite element error estimators and we design a robust adaptive mesh refinement strategy. Numerical tests confirm that errors in different norms scale appropriately. Finally, we demonstrate the effectiveness of our a posteriori indicators in guiding mesh refinement. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_17569 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Error estimates and adaptivity for a least-squares method applied to the Monge-Ampère equation Caboussat, Alexandre Peruso, Anna Picasso, Marco Numerical Analysis 65M60 (Primary) 65M50 (Secondary) G.1.8; G.1.6 We introduce novel a posteriori error indicators for a nonlinear least-squares solver for smooth solutions of the Monge--Ampère equation on convex polygonal domains in $\mathbb{R}^2$. At each iteration, our iterative scheme decouples the problem into (i) a pointwise nonlinear minimization problem and (ii) a linear biharmonic variational problem. For the latter, we derive an equivalence to a biharmonic problem with Navier boundary conditions and solve it via mixed piecewise-linear finite elements. Reformulating this as a coupled second-order system, we derive a priori and a posteriori $\mathbb{P}^1$ finite element error estimators and we design a robust adaptive mesh refinement strategy. Numerical tests confirm that errors in different norms scale appropriately. Finally, we demonstrate the effectiveness of our a posteriori indicators in guiding mesh refinement. |
| title | Error estimates and adaptivity for a least-squares method applied to the Monge-Ampère equation |
| topic | Numerical Analysis 65M60 (Primary) 65M50 (Secondary) G.1.8; G.1.6 |
| url | https://arxiv.org/abs/2507.17569 |