An adaptive least-squares algorithm for the elliptic Monge–Ampère equation
We address the numerical solution of the Dirichlet problem for the two-dimensional elliptic Monge–Ampère equation using a least-squares/relaxation approach. The relaxation algorithm allows the decoupling of the differential operators from the nonlinearities of the equation, within a splitting approa...
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| Language: | English |
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Académie des sciences
2023-10-01
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| Series: | Comptes Rendus. Mécanique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.222/ |
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| author | Caboussat, Alexandre Gourzoulidis, Dimitrios Picasso, Marco |
| author_facet | Caboussat, Alexandre Gourzoulidis, Dimitrios Picasso, Marco |
| author_sort | Caboussat, Alexandre |
| collection | DOAJ |
| description | We address the numerical solution of the Dirichlet problem for the two-dimensional elliptic Monge–Ampère equation using a least-squares/relaxation approach. The relaxation algorithm allows the decoupling of the differential operators from the nonlinearities of the equation, within a splitting approach. The approximation relies on mixed low order finite element methods with regularization techniques. In order to account for data singularities in non-smooth cases, we introduce an adaptive mesh refinement technique. The error indicator is based an independent formulation of the Monge–Ampère equation under divergence form, which allows to explicit a residual term. We show that the error is bounded from above by an a posteriori error indicator plus an extra term that remains to be estimated. This indicator is then used within the existing least-squares framework. The results of numerical experiments support the convergence of our relaxation method to a convex classical solution, if such a solution exists. Otherwise they support convergence to a generalized solution in a least-squares sense. Adaptive mesh refinement proves to be efficient, robust, and accurate to tackle test cases with singularities. |
| format | Article |
| id | doaj-art-7ce8052ed6ed45c3a6cd24f964a943d5 |
| institution | Kabale University |
| issn | 1873-7234 |
| language | English |
| publishDate | 2023-10-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mécanique |
| spelling | doaj-art-7ce8052ed6ed45c3a6cd24f964a943d52025-02-07T13:46:20ZengAcadémie des sciencesComptes Rendus. Mécanique1873-72342023-10-01351S127729210.5802/crmeca.22210.5802/crmeca.222An adaptive least-squares algorithm for the elliptic Monge–Ampère equationCaboussat, Alexandre0https://orcid.org/0000-0003-0964-3603Gourzoulidis, Dimitrios1https://orcid.org/0000-0002-3477-3226Picasso, Marco2https://orcid.org/0000-0002-0069-5856Geneva School of Business Administration (HEG-GE), University of Applied Sciences and Arts Western Switzerland (HES-SO), 1227 Carouge, Geneva, SwitzerlandGeneva School of Business Administration (HEG-GE), University of Applied Sciences and Arts Western Switzerland (HES-SO), 1227 Carouge, Geneva, Switzerland; Institute of Mathematics, Ecole Polytechnique Fédérale de Lausanne (EPFL) 1018 Lausanne, SwitzerlandInstitute of Mathematics, Ecole Polytechnique Fédérale de Lausanne (EPFL) 1018 Lausanne, SwitzerlandWe address the numerical solution of the Dirichlet problem for the two-dimensional elliptic Monge–Ampère equation using a least-squares/relaxation approach. The relaxation algorithm allows the decoupling of the differential operators from the nonlinearities of the equation, within a splitting approach. The approximation relies on mixed low order finite element methods with regularization techniques. In order to account for data singularities in non-smooth cases, we introduce an adaptive mesh refinement technique. The error indicator is based an independent formulation of the Monge–Ampère equation under divergence form, which allows to explicit a residual term. We show that the error is bounded from above by an a posteriori error indicator plus an extra term that remains to be estimated. This indicator is then used within the existing least-squares framework. The results of numerical experiments support the convergence of our relaxation method to a convex classical solution, if such a solution exists. Otherwise they support convergence to a generalized solution in a least-squares sense. Adaptive mesh refinement proves to be efficient, robust, and accurate to tackle test cases with singularities.https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.222/Fully nonlinear PDEMonge–Ampère equationLeast-squares algorithmMixed finite elementsAdaptive mesh refinementNon-smooth problems |
| spellingShingle | Caboussat, Alexandre Gourzoulidis, Dimitrios Picasso, Marco An adaptive least-squares algorithm for the elliptic Monge–Ampère equation Comptes Rendus. Mécanique Fully nonlinear PDE Monge–Ampère equation Least-squares algorithm Mixed finite elements Adaptive mesh refinement Non-smooth problems |
| title | An adaptive least-squares algorithm for the elliptic Monge–Ampère equation |
| title_full | An adaptive least-squares algorithm for the elliptic Monge–Ampère equation |
| title_fullStr | An adaptive least-squares algorithm for the elliptic Monge–Ampère equation |
| title_full_unstemmed | An adaptive least-squares algorithm for the elliptic Monge–Ampère equation |
| title_short | An adaptive least-squares algorithm for the elliptic Monge–Ampère equation |
| title_sort | adaptive least squares algorithm for the elliptic monge ampere equation |
| topic | Fully nonlinear PDE Monge–Ampère equation Least-squares algorithm Mixed finite elements Adaptive mesh refinement Non-smooth problems |
| url | https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.222/ |
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