Nonconforming virtual element method for the Monge-Ampère equation
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866917433499975680 |
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| author | Congreve, Scott Hodson, Alice Pradhan, Anwesh |
| author_facet | Congreve, Scott Hodson, Alice Pradhan, Anwesh |
| contents | In this article, we develop the $C^1$-nonconforming $C^0$-conforming virtual element method (VEM) for the vanishing moment approximation of the second-order fully nonlinear Monge-Ampère equation in two dimensions. In the vanishing moment equation an artificial biharmonic term is introduced which produces a quasilinear fourth order problem. We derive optimal a priori error estimates in the $H^2$-, $H^1$- and $L^2$-norms for the virtual element method, and show the existence and uniqueness of the virtual element solution. We perform several numerical experiments to validate the convergence rate of the error with respect to the mesh size. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_22589 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Nonconforming virtual element method for the Monge-Ampère equation Congreve, Scott Hodson, Alice Pradhan, Anwesh Numerical Analysis 65N30 (Primary) 35J96, 65N15 (Secondary) In this article, we develop the $C^1$-nonconforming $C^0$-conforming virtual element method (VEM) for the vanishing moment approximation of the second-order fully nonlinear Monge-Ampère equation in two dimensions. In the vanishing moment equation an artificial biharmonic term is introduced which produces a quasilinear fourth order problem. We derive optimal a priori error estimates in the $H^2$-, $H^1$- and $L^2$-norms for the virtual element method, and show the existence and uniqueness of the virtual element solution. We perform several numerical experiments to validate the convergence rate of the error with respect to the mesh size. |
| title | Nonconforming virtual element method for the Monge-Ampère equation |
| topic | Numerical Analysis 65N30 (Primary) 35J96, 65N15 (Secondary) |
| url | https://arxiv.org/abs/2604.22589 |