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| Format: | Preprint |
| Veröffentlicht: |
2026
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| Online-Zugang: | https://arxiv.org/abs/2603.23167 |
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| _version_ | 1866914417523818496 |
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| author | Qi, Ruisheng Wang, Xiaojie |
| author_facet | Qi, Ruisheng Wang, Xiaojie |
| contents | The present paper proposes new fully discrete schemes for long-time approximations of stochastic partial differential equations (SPDEs) with non-globally Lipschitz coefficients in a bounded domain $D \subset \R^d, d =1,2,3 $. A novel family of linearly implicit time-stepping schemes is introduced, based on a standard Galerkin finite element spatial semi-discretization. A distinguishing feature of the schemes is that the proposed finite element fully discrete approximations preserve uniform-in-time moment bounds in a Banach space $L^{r}(D), r >2$, without requiring any restriction on the time-space discretization stepsize ratio. %established... To show it, some non-standard arguments are developed. First, we derive long-time error estimates in the Banach space $L^r(D)$ for finite element fully discrete approximations of the deterministic linear parabolic equation with non-smooth initial value, which is, to our knowledge, new for the literature on numerical PDEs and of independent interest. These error estimates together with the contractive property of the semi-group in $L^{r}(D), r > 2$, the dissipativity of the nonlinearity and the particular benefit of the taming strategy help us establish the desired uniform-in-time moment bounds. Then both strong and weak error bounds of the proposed schemes are carefully analyzed in a setting of low regularity, with uniform-in-time convergence rates obtained for cases of both space-time white and trace-class noises. The analysis is highly nontrivial, due to the finite element discretization, the low regularity and the presence of the super-linearly growing nonlinearity. %in the long-time scenario... the discretization parameters $h$ and $τ$. Finally, numerical results are presented to verify the previous theoretical findings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_23167 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Long-time error analysis of finite element fully discrete schemes for SPDEs with non-globally Lipschitz coefficients Qi, Ruisheng Wang, Xiaojie Numerical Analysis The present paper proposes new fully discrete schemes for long-time approximations of stochastic partial differential equations (SPDEs) with non-globally Lipschitz coefficients in a bounded domain $D \subset \R^d, d =1,2,3 $. A novel family of linearly implicit time-stepping schemes is introduced, based on a standard Galerkin finite element spatial semi-discretization. A distinguishing feature of the schemes is that the proposed finite element fully discrete approximations preserve uniform-in-time moment bounds in a Banach space $L^{r}(D), r >2$, without requiring any restriction on the time-space discretization stepsize ratio. %established... To show it, some non-standard arguments are developed. First, we derive long-time error estimates in the Banach space $L^r(D)$ for finite element fully discrete approximations of the deterministic linear parabolic equation with non-smooth initial value, which is, to our knowledge, new for the literature on numerical PDEs and of independent interest. These error estimates together with the contractive property of the semi-group in $L^{r}(D), r > 2$, the dissipativity of the nonlinearity and the particular benefit of the taming strategy help us establish the desired uniform-in-time moment bounds. Then both strong and weak error bounds of the proposed schemes are carefully analyzed in a setting of low regularity, with uniform-in-time convergence rates obtained for cases of both space-time white and trace-class noises. The analysis is highly nontrivial, due to the finite element discretization, the low regularity and the presence of the super-linearly growing nonlinearity. %in the long-time scenario... the discretization parameters $h$ and $τ$. Finally, numerical results are presented to verify the previous theoretical findings. |
| title | Long-time error analysis of finite element fully discrete schemes for SPDEs with non-globally Lipschitz coefficients |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2603.23167 |