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Volume 42, Issue 3
Sharp Pointwise-in-Time Error Estimate of L1 Scheme for Nonlinear Subdiffusion Equations

Dongfang Li, Hongyu Qin & Jiwei Zhang

DOI: 10.4208/jcm.2205-m2021-0316

J. Comp. Math., 42 (2024), pp. 662-678.

Published online: 2024-04

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  • Abstract

An essential feature of the subdiffusion equations with the $α$-order time fractional derivative is the weak singularity at the initial time. The weak regularity of the solution is usually characterized by a regularity parameter $σ ∈ (0, 1) ∪ (1, 2).$ Under this general regularity assumption, we present a rigorous analysis for the truncation errors and develop a new tool to obtain the stability results, i.e., a refined discrete fractional-type Grönwall inequality (DFGI). After that, we obtain the pointwise-in-time error estimate of the widely used L1 scheme for nonlinear subdiffusion equations. The present results fill the gap on some interesting convergence results of L1 scheme on $σ ∈ (0, α) ∪ (α, 1) ∪ (1, 2].$ Numerical experiments are provided to demonstrate the effectiveness of our theoretical analysis.

  • Keywords

Sharp pointwise-in-time error estimate, L1 scheme, Nonlinear subdiffusion equations, Non-smooth solutions.

  • AMS Subject Headings

35R11, 65M12, 65M60

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-42-662, author = {Li , DongfangQin , Hongyu and Zhang , Jiwei}, title = {Sharp Pointwise-in-Time Error Estimate of L1 Scheme for Nonlinear Subdiffusion Equations}, journal = {Journal of Computational Mathematics}, year = {2024}, volume = {42}, number = {3}, pages = {662--678}, abstract = {

An essential feature of the subdiffusion equations with the $α$-order time fractional derivative is the weak singularity at the initial time. The weak regularity of the solution is usually characterized by a regularity parameter $σ ∈ (0, 1) ∪ (1, 2).$ Under this general regularity assumption, we present a rigorous analysis for the truncation errors and develop a new tool to obtain the stability results, i.e., a refined discrete fractional-type Grönwall inequality (DFGI). After that, we obtain the pointwise-in-time error estimate of the widely used L1 scheme for nonlinear subdiffusion equations. The present results fill the gap on some interesting convergence results of L1 scheme on $σ ∈ (0, α) ∪ (α, 1) ∪ (1, 2].$ Numerical experiments are provided to demonstrate the effectiveness of our theoretical analysis.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2205-m2021-0316}, url = {http://global-sci.org/intro/article_detail/jcm/23031.html} }
TY - JOUR T1 - Sharp Pointwise-in-Time Error Estimate of L1 Scheme for Nonlinear Subdiffusion Equations AU - Li , Dongfang AU - Qin , Hongyu AU - Zhang , Jiwei JO - Journal of Computational Mathematics VL - 3 SP - 662 EP - 678 PY - 2024 DA - 2024/04 SN - 42 DO - http://doi.org/10.4208/jcm.2205-m2021-0316 UR - https://global-sci.org/intro/article_detail/jcm/23031.html KW - Sharp pointwise-in-time error estimate, L1 scheme, Nonlinear subdiffusion equations, Non-smooth solutions. AB -

An essential feature of the subdiffusion equations with the $α$-order time fractional derivative is the weak singularity at the initial time. The weak regularity of the solution is usually characterized by a regularity parameter $σ ∈ (0, 1) ∪ (1, 2).$ Under this general regularity assumption, we present a rigorous analysis for the truncation errors and develop a new tool to obtain the stability results, i.e., a refined discrete fractional-type Grönwall inequality (DFGI). After that, we obtain the pointwise-in-time error estimate of the widely used L1 scheme for nonlinear subdiffusion equations. The present results fill the gap on some interesting convergence results of L1 scheme on $σ ∈ (0, α) ∪ (α, 1) ∪ (1, 2].$ Numerical experiments are provided to demonstrate the effectiveness of our theoretical analysis.

Dongfang Li, Hongyu Qin & Jiwei Zhang. (2024). Sharp Pointwise-in-Time Error Estimate of L1 Scheme for Nonlinear Subdiffusion Equations. Journal of Computational Mathematics. 42 (3). 662-678. doi:10.4208/jcm.2205-m2021-0316
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