On the relation between Galerkin approximations and canonical best-approximations of solutions to some non-linear Schrödinger equations
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| Format: | Preprint |
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2025
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| _version_ | 1866929710384021504 |
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| author | Hassan, Muhammad Maday, Yvon Wang, Yipeng |
| author_facet | Hassan, Muhammad Maday, Yvon Wang, Yipeng |
| contents | In this paper, we establish a superconvergence property of Galerkin approximations to some non-linear Schrödinger equations of Gross-Pitaevskii type. More precisely, denoting by $u^*\in X \subseteq H^1(Ω)$ the exact solution to such an equation, by $\{X_δ\}_{δ>0}$, a sequence of conforming subspaces of $X$ satisfying the approximation property, by $u_δ^*\in X_δ$ the Galerkin solution to the equation, and by $Π^X_δ u^*$, the $(\cdot, \cdot)_{X}$-best approximation in $X_δ$ of $u^*$, we show -- under some assumptions -- that $u_δ^*$ converges at a higher rate to $Π^X_δ u^*$ than to $u^*$ in both the $L^2$ norm and the canonical $H^1$ norm. Our results apply to conforming finite element discretisations as well as spectral Galerkin methods based on polynomials or Fourier (plane-wave) expansions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_07638 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the relation between Galerkin approximations and canonical best-approximations of solutions to some non-linear Schrödinger equations Hassan, Muhammad Maday, Yvon Wang, Yipeng Numerical Analysis 65N15, 65N25, 65N30, 65N35 In this paper, we establish a superconvergence property of Galerkin approximations to some non-linear Schrödinger equations of Gross-Pitaevskii type. More precisely, denoting by $u^*\in X \subseteq H^1(Ω)$ the exact solution to such an equation, by $\{X_δ\}_{δ>0}$, a sequence of conforming subspaces of $X$ satisfying the approximation property, by $u_δ^*\in X_δ$ the Galerkin solution to the equation, and by $Π^X_δ u^*$, the $(\cdot, \cdot)_{X}$-best approximation in $X_δ$ of $u^*$, we show -- under some assumptions -- that $u_δ^*$ converges at a higher rate to $Π^X_δ u^*$ than to $u^*$ in both the $L^2$ norm and the canonical $H^1$ norm. Our results apply to conforming finite element discretisations as well as spectral Galerkin methods based on polynomials or Fourier (plane-wave) expansions. |
| title | On the relation between Galerkin approximations and canonical best-approximations of solutions to some non-linear Schrödinger equations |
| topic | Numerical Analysis 65N15, 65N25, 65N30, 65N35 |
| url | https://arxiv.org/abs/2502.07638 |