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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.05637 |
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| _version_ | 1866911654874185728 |
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| author | Hu, Qiya Luo, Yuhan |
| author_facet | Hu, Qiya Luo, Yuhan |
| contents | It is known that the weighted $L^2$ projection operator exhibits approximation properties different from those of the classical $L^2$ projection, in the sense that the $L^2$ error of the weighted $L^2$ projection of an $H^1$ function generally cannot be bounded by the $H^1$ semi-norm of the function. In this paper, we establish sharper $L^2$ error estimates for the weighted $L^2$ projection of an $H^1$ function under general weight distributions. These new estimates show that the $L^2$ errors of the weighted $L^2$ projection can be controlled by the $H^1$ semi-norm of the function, except when the weight distribution is highly irregular, such as those resembling a ``checkerboard" pattern. These results can be applied to more refined analyses of domain decomposition methods and multigrid methods for certain partial differential equations with large jump coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_05637 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | New error estimates of the weighted $L^2$ projections Hu, Qiya Luo, Yuhan Numerical Analysis It is known that the weighted $L^2$ projection operator exhibits approximation properties different from those of the classical $L^2$ projection, in the sense that the $L^2$ error of the weighted $L^2$ projection of an $H^1$ function generally cannot be bounded by the $H^1$ semi-norm of the function. In this paper, we establish sharper $L^2$ error estimates for the weighted $L^2$ projection of an $H^1$ function under general weight distributions. These new estimates show that the $L^2$ errors of the weighted $L^2$ projection can be controlled by the $H^1$ semi-norm of the function, except when the weight distribution is highly irregular, such as those resembling a ``checkerboard" pattern. These results can be applied to more refined analyses of domain decomposition methods and multigrid methods for certain partial differential equations with large jump coefficients. |
| title | New error estimates of the weighted $L^2$ projections |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2605.05637 |