论文标题
长时间$ h^1 $ - 快速l2-1 $_σ$方法的稳定性，用于子扩散方程
Long time $H^1$-stability of fast L2-1$_σ$ method on general nonuniform meshes for subdiffusion equations
论文作者
论文摘要
在这项工作中，对一般非均匀网格的快速L2-1 $_σ$方法的全局时间$ h^1 $ - 稳定性进行了细分方程，其中Caputo分数衍生物中的卷积内核通过指数的总和近似。在某些轻度限制时间对时间步长下，证明与快速L2-1 $_σ$公式相关的双线性形式一直被证明是有史以来的半半决赛。结果，可以在线性和半线性次散开方程中得出快速L2-1 $_σ$方案的均匀全球$ H^1 $ - 稳定性，因为$ h^1 $ - norm均匀地界定为时间，因为时间趋于无限。据我们所知，这似乎是全局$ h^1 $ h^1 $ - 稳定性的第一项工作，快速l2-1 $_σ$方案在一般不均匀的网格上用于子扩散方程。此外，基于对系数的更微妙的分析，对快速L2-1 $_σ$方案的急剧有限时间$ h^1 $ -ERROR估算值，与现有作品相比，对时间步长的限制放宽了。
In this work, the global-in-time $H^1$-stability of a fast L2-1$_σ$ method on general nonuniform meshes is studied for subdiffusion equations, where the convolution kernel in the Caputo fractional derivative is approximated by sum of exponentials. Under some mild restrictions on time stepsize, a bilinear form associated with the fast L2-1$_σ$ formula is proved to be positive semidefinite for all time. As a consequence, the uniform global-in-time $H^1$-stability of the fast L2-1$_σ$ schemes can be derived for both linear and semilinear subdiffusion equations, in the sense that the $H^1$-norm is uniformly bounded as the time tends to infinity. To the best of our knowledge, this appears to be the first work for the global-in-time $H^1$-stability of fast L2-1$_σ$ scheme on general nonuniform meshes for subdiffusion equations. Moreover, the sharp finite time $H^1$-error estimate for the fast L2-1$_σ$ schemes is reproved based on more delicate analysis of coefficients where the restriction on time step ratios is relaxed comparing to existing works.
