A quasi-interpolation operator yielding fully computable error bounds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915392042041344 |
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| author | Chaumont-Frelet, T. Vohralik, M. |
| author_facet | Chaumont-Frelet, T. Vohralik, M. |
| contents | We design a quasi-interpolation operator from the Sobolev space $H^1_0(Ω)$ to its finite-dimensional finite element subspace formed by piecewise polynomials on a simplicial mesh with a computable approximation constant. The operator 1) is defined on the entire $H^1_0(Ω)$, no additional regularity is needed; 2) allows for an arbitrary polynomial degree; 3) works in any space dimension; 4) is defined locally, in vertex patches of mesh elements; 5) yields optimal estimates for both the $H^1$ seminorm and the $L^2$ norm error; 6) gives a computable constant for both the $H^1$ seminorm and the $L^2$ norm error; 7) leads to the equivalence of global-best and local-best errors; 8) possesses the projection property. Its construction follows the so-called potential reconstruction from a posteriori error analysis. Numerical experiments illustrate that our quasi-interpolation operator systematically gives the correct convergence rates in both the $H^1$ seminorm and the $L^2$ norm and its certified overestimation factor is rather sharp and stable in all tested situations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_11819 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A quasi-interpolation operator yielding fully computable error bounds Chaumont-Frelet, T. Vohralik, M. Numerical Analysis We design a quasi-interpolation operator from the Sobolev space $H^1_0(Ω)$ to its finite-dimensional finite element subspace formed by piecewise polynomials on a simplicial mesh with a computable approximation constant. The operator 1) is defined on the entire $H^1_0(Ω)$, no additional regularity is needed; 2) allows for an arbitrary polynomial degree; 3) works in any space dimension; 4) is defined locally, in vertex patches of mesh elements; 5) yields optimal estimates for both the $H^1$ seminorm and the $L^2$ norm error; 6) gives a computable constant for both the $H^1$ seminorm and the $L^2$ norm error; 7) leads to the equivalence of global-best and local-best errors; 8) possesses the projection property. Its construction follows the so-called potential reconstruction from a posteriori error analysis. Numerical experiments illustrate that our quasi-interpolation operator systematically gives the correct convergence rates in both the $H^1$ seminorm and the $L^2$ norm and its certified overestimation factor is rather sharp and stable in all tested situations. |
| title | A quasi-interpolation operator yielding fully computable error bounds |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2507.11819 |