Optimal-order finite difference approximation of generalized solutions to the biharmonic equation in a cube
We prove an optimal-order error bound in the discrete $H^2(\Omega)$ norm for finite difference approximations of the first boundary-value problem for the biharmonic equation in $n$ space dimensions, with $n \in \{2,\dots,7\}$, whose generalized solution belongs to the Sobolev space $H^s(\Omega) \cap...
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| Format: | Journal article |
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Society for Industrial and Applied Mathematics
2020
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| _version_ | 1797086067856244736 |
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| author | Müller, S Schweiger, F Süli, E |
| author_facet | Müller, S Schweiger, F Süli, E |
| author_sort | Müller, S |
| collection | OXFORD |
| description | We prove an optimal-order error bound in the discrete $H^2(\Omega)$ norm for finite difference approximations of the first boundary-value problem for the biharmonic equation in $n$ space dimensions, with $n \in \{2,\dots,7\}$, whose generalized solution belongs to the Sobolev space $H^s(\Omega) \cap H^2_0(\Omega)$ for $\frac{1}{2} \max(5,n) < s \leq 4$, where $\Omega = (0,1)^n$. The result extends the range of the Sobolev index $s$ in the best convergence results currently available in the literature to the maximal range admitted by the Sobolev embedding of $H^s(\Omega)$ into $C(\overline\Omega)$ in $n$ space dimensions. |
| first_indexed | 2024-03-07T02:16:45Z |
| format | Journal article |
| id | oxford-uuid:a283a3dd-3da4-4f6a-87cf-89437a647840 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T02:16:45Z |
| publishDate | 2020 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | oxford-uuid:a283a3dd-3da4-4f6a-87cf-89437a6478402022-03-27T02:20:38ZOptimal-order finite difference approximation of generalized solutions to the biharmonic equation in a cube Journal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:a283a3dd-3da4-4f6a-87cf-89437a647840Symplectic Elements at OxfordSociety for Industrial and Applied Mathematics2020Müller, SSchweiger, FSüli, EWe prove an optimal-order error bound in the discrete $H^2(\Omega)$ norm for finite difference approximations of the first boundary-value problem for the biharmonic equation in $n$ space dimensions, with $n \in \{2,\dots,7\}$, whose generalized solution belongs to the Sobolev space $H^s(\Omega) \cap H^2_0(\Omega)$ for $\frac{1}{2} \max(5,n) < s \leq 4$, where $\Omega = (0,1)^n$. The result extends the range of the Sobolev index $s$ in the best convergence results currently available in the literature to the maximal range admitted by the Sobolev embedding of $H^s(\Omega)$ into $C(\overline\Omega)$ in $n$ space dimensions. |
| spellingShingle | Müller, S Schweiger, F Süli, E Optimal-order finite difference approximation of generalized solutions to the biharmonic equation in a cube |
| title | Optimal-order finite difference approximation of generalized solutions to the biharmonic equation in a cube |
| title_full | Optimal-order finite difference approximation of generalized solutions to the biharmonic equation in a cube |
| title_fullStr | Optimal-order finite difference approximation of generalized solutions to the biharmonic equation in a cube |
| title_full_unstemmed | Optimal-order finite difference approximation of generalized solutions to the biharmonic equation in a cube |
| title_short | Optimal-order finite difference approximation of generalized solutions to the biharmonic equation in a cube |
| title_sort | optimal order finite difference approximation of generalized solutions to the biharmonic equation in a cube |
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