Low-order finite element complex with application to a fourth-order elliptic singular perturbation problem
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909942295822336 |
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| author | Cui, Xuewei Huang, Xuehai |
| author_facet | Cui, Xuewei Huang, Xuehai |
| contents | A low-order nonconforming finite element discretization of a smooth de Rham complex starting from the $H^2$ space in three dimensions is proposed, involving an $H^2$-nonconforming finite element space, a new tangentially continuous $H^1$-nonconforming vector-valued finite element space, the lowest-order Raviart-Thomas space, and piecewise constant functions. While nonconforming for the smooth complex, the discretization conforms to the classical de Rham complex. It is applied to develop a decoupled mixed finite element method for a fourth-order elliptic singular perturbation problem, focusing on the discretization of a generalized singularly perturbed Stokes-type equation. In contrast to Nitsche's method, which requires additional stabilization to handle boundary layers, the nodal interpolation operator for the lowest-order Nédélec element of the second kind is introduced into the discrete bilinear forms. This modification yields a decoupled mixed method that achieves optimal convergence rates uniformly with respect to the perturbation parameter, even in the presence of strong boundary layers, without requiring any additional stabilization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_20240 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Low-order finite element complex with application to a fourth-order elliptic singular perturbation problem Cui, Xuewei Huang, Xuehai Numerical Analysis A low-order nonconforming finite element discretization of a smooth de Rham complex starting from the $H^2$ space in three dimensions is proposed, involving an $H^2$-nonconforming finite element space, a new tangentially continuous $H^1$-nonconforming vector-valued finite element space, the lowest-order Raviart-Thomas space, and piecewise constant functions. While nonconforming for the smooth complex, the discretization conforms to the classical de Rham complex. It is applied to develop a decoupled mixed finite element method for a fourth-order elliptic singular perturbation problem, focusing on the discretization of a generalized singularly perturbed Stokes-type equation. In contrast to Nitsche's method, which requires additional stabilization to handle boundary layers, the nodal interpolation operator for the lowest-order Nédélec element of the second kind is introduced into the discrete bilinear forms. This modification yields a decoupled mixed method that achieves optimal convergence rates uniformly with respect to the perturbation parameter, even in the presence of strong boundary layers, without requiring any additional stabilization. |
| title | Low-order finite element complex with application to a fourth-order elliptic singular perturbation problem |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2506.20240 |