Nonconforming virtual element methods for fourth-order nonlinear reaction-diffusion systems: a unified framework and analysis
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917275653636096 |
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| author | Adak, Dibyendu Mora, David Silgado, Alberth |
| author_facet | Adak, Dibyendu Mora, David Silgado, Alberth |
| contents | We develop a unified framework for the design and analysis of high-order nonconforming virtual element methods for nonlinear fourth-order reaction--diffusion problems in two dimensions, with emphasis on clamped, Navier, and Cahn--Hilliard-type boundary conditions. Time discretization is performed using the backward Euler scheme, while the spatial approximation relies on nonconforming virtual element spaces of arbitrary order $k \ge 2$, encompassing both $C^0$-nonconforming and Morley-type methods. A key contribution of this work is the development of a novel and rigorous unified error analysis for these numerical schemes, applicable to domains that are not necessarily convex, differing from the existing literature. By introducing a class of Companion operators, we construct novel Ritz-type projections and derive a new error equation that enables us to obtain optimal error estimates for the scheme under a minimal spatial regularity assumption on the weak solution. Finally, we present numerical experiments on polygonal meshes as applications of the proposed framework, including the extended Fisher--Kolmogorov equation, and a fourth-order model with Cahn--Hilliard-type boundary conditions, which validate the theoretical results and illustrate the performance of the method for the three classes of boundary conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14309 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Nonconforming virtual element methods for fourth-order nonlinear reaction-diffusion systems: a unified framework and analysis Adak, Dibyendu Mora, David Silgado, Alberth Numerical Analysis 65N30, 65M15, 35J35 We develop a unified framework for the design and analysis of high-order nonconforming virtual element methods for nonlinear fourth-order reaction--diffusion problems in two dimensions, with emphasis on clamped, Navier, and Cahn--Hilliard-type boundary conditions. Time discretization is performed using the backward Euler scheme, while the spatial approximation relies on nonconforming virtual element spaces of arbitrary order $k \ge 2$, encompassing both $C^0$-nonconforming and Morley-type methods. A key contribution of this work is the development of a novel and rigorous unified error analysis for these numerical schemes, applicable to domains that are not necessarily convex, differing from the existing literature. By introducing a class of Companion operators, we construct novel Ritz-type projections and derive a new error equation that enables us to obtain optimal error estimates for the scheme under a minimal spatial regularity assumption on the weak solution. Finally, we present numerical experiments on polygonal meshes as applications of the proposed framework, including the extended Fisher--Kolmogorov equation, and a fourth-order model with Cahn--Hilliard-type boundary conditions, which validate the theoretical results and illustrate the performance of the method for the three classes of boundary conditions. |
| title | Nonconforming virtual element methods for fourth-order nonlinear reaction-diffusion systems: a unified framework and analysis |
| topic | Numerical Analysis 65N30, 65M15, 35J35 |
| url | https://arxiv.org/abs/2602.14309 |