Symmetric-Tensor Distributional Mixed Method for Fourth-Order Elliptic Singular Perturbation Problem
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917555301515264 |
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| author | Huang, Xuehai Zhao, Xinyue |
| author_facet | Huang, Xuehai Zhao, Xinyue |
| contents | A symmetric-tensor distributional mixed method for a fourth-order elliptic singular perturbation problem is developed in this paper. The moment variable is approximated by normal-normal continuous symmetric tensor elements, while the scalar variable is represented by an $H^1$-nonconforming virtual element space coupled with a polynomial multiplier on interior subsimplices of codimension two. Optimal parameter-uniform error estimates are derived, independent of the presence of boundary layers. A hybridized form of the method is also equivalent to stabilization-free weak Galerkin and $H^2$-nonconforming virtual element methods. In two dimensions, a close connection of the distributional mixed method to the classical Hellan-Herrmann-Johnson (HHJ) method is established, by naturally identifying the scalar virtual element-multiplier pair with the Lagrange finite element space. Thus the proposed method extends the two-dimensional HHJ method to arbitrary spatial dimensions. Three-dimensional numerical experiments support the theoretical convergence and robustness estimates. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2606_02188 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Symmetric-Tensor Distributional Mixed Method for Fourth-Order Elliptic Singular Perturbation Problem Huang, Xuehai Zhao, Xinyue Numerical Analysis 65N12, 65N22, 65N30, 15A69 A symmetric-tensor distributional mixed method for a fourth-order elliptic singular perturbation problem is developed in this paper. The moment variable is approximated by normal-normal continuous symmetric tensor elements, while the scalar variable is represented by an $H^1$-nonconforming virtual element space coupled with a polynomial multiplier on interior subsimplices of codimension two. Optimal parameter-uniform error estimates are derived, independent of the presence of boundary layers. A hybridized form of the method is also equivalent to stabilization-free weak Galerkin and $H^2$-nonconforming virtual element methods. In two dimensions, a close connection of the distributional mixed method to the classical Hellan-Herrmann-Johnson (HHJ) method is established, by naturally identifying the scalar virtual element-multiplier pair with the Lagrange finite element space. Thus the proposed method extends the two-dimensional HHJ method to arbitrary spatial dimensions. Three-dimensional numerical experiments support the theoretical convergence and robustness estimates. |
| title | Symmetric-Tensor Distributional Mixed Method for Fourth-Order Elliptic Singular Perturbation Problem |
| topic | Numerical Analysis 65N12, 65N22, 65N30, 15A69 |
| url | https://arxiv.org/abs/2606.02188 |