Uniform error analysis of a rectangular Morley finite element method on a Shishkin mesh for a 4th-order singularly perturbed boundary value problem
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866911127855693824 |
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| author | Meng, Xiangyun Stynes, Martin |
| author_facet | Meng, Xiangyun Stynes, Martin |
| contents | The singularly perturbed reaction-diffusion problem $\varepsilon^2Δ^2 u - \mathrm{div}\left(c\nabla u\right) = f$ is considered on the unit square $Ω$ in $\mathbb{R}^2$ with homogenous Dirichlet boundary conditions. Its solution typically contains boundary layers on all sides of~$Ω$. It is discretised by a finite element method that uses rectangular Morley elements on a Shishkin mesh. In an associated energy-type norm that is natural for this problem, we prove an $O(\varepsilon^{1/2}N^{-1}+\varepsilon N^{-1}\ln N + N^{-3/2})$ rate of convergence for the error in the computed solution, where $N$~is the number of mesh intervals in each coordinate direction. Thus in the most troublesome regime when $\varepsilon \approx N^{-1}$, our method is proved to attain an $O(N^{-3/2})$ rate of convergence, which is shown to be sharp by our numerical experiments and is superior to the $O(N^{-1/2})$ rate that is proved in Meng & Stynes, Adv. Comput. Math. 2019 when Adini finite elements are used to solve the same problem on the same mesh. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_20857 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniform error analysis of a rectangular Morley finite element method on a Shishkin mesh for a 4th-order singularly perturbed boundary value problem Meng, Xiangyun Stynes, Martin Numerical Analysis The singularly perturbed reaction-diffusion problem $\varepsilon^2Δ^2 u - \mathrm{div}\left(c\nabla u\right) = f$ is considered on the unit square $Ω$ in $\mathbb{R}^2$ with homogenous Dirichlet boundary conditions. Its solution typically contains boundary layers on all sides of~$Ω$. It is discretised by a finite element method that uses rectangular Morley elements on a Shishkin mesh. In an associated energy-type norm that is natural for this problem, we prove an $O(\varepsilon^{1/2}N^{-1}+\varepsilon N^{-1}\ln N + N^{-3/2})$ rate of convergence for the error in the computed solution, where $N$~is the number of mesh intervals in each coordinate direction. Thus in the most troublesome regime when $\varepsilon \approx N^{-1}$, our method is proved to attain an $O(N^{-3/2})$ rate of convergence, which is shown to be sharp by our numerical experiments and is superior to the $O(N^{-1/2})$ rate that is proved in Meng & Stynes, Adv. Comput. Math. 2019 when Adini finite elements are used to solve the same problem on the same mesh. |
| title | Uniform error analysis of a rectangular Morley finite element method on a Shishkin mesh for a 4th-order singularly perturbed boundary value problem |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2508.20857 |