Local convergence analysis of L1/finite element scheme for a constant delay reaction-subdiffusion equation with uniform time mesh
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| Format: | Preprint |
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2024
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| _version_ | 1866909244466397184 |
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| author | Bu, Weiping Zheng, Xin |
| author_facet | Bu, Weiping Zheng, Xin |
| contents | The aim of this paper is to develop a refined error estimate of L1/finite element scheme for a reaction-subdiffusion equation with constant delay $τ$ and uniform time mesh. Under the non-uniform multi-singularity assumption of exact solution in time, the local truncation errors of the L1 scheme with uniform mesh is investigated. Then we introduce a fully discrete finite element scheme of the considered problem. Next, a novel discrete fractional Grönwall inequality with constant delay term is proposed, which does not include the increasing Mittag-Leffler function comparing with some popular other cases. By applying this Grönwall inequality, we obtain the pointwise-in-time and piecewise-in-time error estimates of the finite element scheme without the Mittag-Leffler function. In particular, the latter shows that, for the considered interval $((i-1)τ,iτ]$, although the convergence in time is low for $i=1$, it will be improved as the increasing $i$, which is consistent with the factual assumption that the smoothness of the solution will be improved as the increasing $i$. Finally, we present some numerical tests to verify the developed theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_05041 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local convergence analysis of L1/finite element scheme for a constant delay reaction-subdiffusion equation with uniform time mesh Bu, Weiping Zheng, Xin Numerical Analysis The aim of this paper is to develop a refined error estimate of L1/finite element scheme for a reaction-subdiffusion equation with constant delay $τ$ and uniform time mesh. Under the non-uniform multi-singularity assumption of exact solution in time, the local truncation errors of the L1 scheme with uniform mesh is investigated. Then we introduce a fully discrete finite element scheme of the considered problem. Next, a novel discrete fractional Grönwall inequality with constant delay term is proposed, which does not include the increasing Mittag-Leffler function comparing with some popular other cases. By applying this Grönwall inequality, we obtain the pointwise-in-time and piecewise-in-time error estimates of the finite element scheme without the Mittag-Leffler function. In particular, the latter shows that, for the considered interval $((i-1)τ,iτ]$, although the convergence in time is low for $i=1$, it will be improved as the increasing $i$, which is consistent with the factual assumption that the smoothness of the solution will be improved as the increasing $i$. Finally, we present some numerical tests to verify the developed theory. |
| title | Local convergence analysis of L1/finite element scheme for a constant delay reaction-subdiffusion equation with uniform time mesh |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2407.05041 |