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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2512.14219 |
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| _version_ | 1866915960265375744 |
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| author | Qiu, Weifeng |
| author_facet | Qiu, Weifeng |
| contents | We propose one finite element method for both second order linear uniformly elliptic PDE in non-divergence form and the uniformly elliptic Hamilton-Jacobi-Bellman (HJB) equation. For both linear elliptic PDE in non-divergence form and the HJB equation, we prove the well-posedness of strong solution in $W^{2,p}(Ω)$ and optimal convergence in discrete $W^{2,p}$-norm of the finite element approximation to the strong solution for $1<p\leq 2$ on convex polyhedra in $\mathbb{R}^{d}$ ($d=2,3$). If the domain is a two dimensional non-convex polygon, $p$ is valid in a more restricted region. Furthermore, we relax the assumptions on the continuity of coefficients of the HJB equation, which have been widely used in literature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_14219 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Analysis of a finite element method for second order uniformly elliptic PDEs in non-divergence form Qiu, Weifeng Numerical Analysis We propose one finite element method for both second order linear uniformly elliptic PDE in non-divergence form and the uniformly elliptic Hamilton-Jacobi-Bellman (HJB) equation. For both linear elliptic PDE in non-divergence form and the HJB equation, we prove the well-posedness of strong solution in $W^{2,p}(Ω)$ and optimal convergence in discrete $W^{2,p}$-norm of the finite element approximation to the strong solution for $1<p\leq 2$ on convex polyhedra in $\mathbb{R}^{d}$ ($d=2,3$). If the domain is a two dimensional non-convex polygon, $p$ is valid in a more restricted region. Furthermore, we relax the assumptions on the continuity of coefficients of the HJB equation, which have been widely used in literature. |
| title | Analysis of a finite element method for second order uniformly elliptic PDEs in non-divergence form |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2512.14219 |