论文标题
$ h^1 $ -NORM稳定性和在subdiffusion方程式上的L2型方法的收敛性和收敛性
$H^1$-norm stability and convergence of an L2-type method on nonuniform meshes for subdiffusion equation
论文作者
论文摘要
当应用于子扩散方程时，这项工作建立了$ h^1 $ norm稳定性和一般不均匀网格的收敛性。在对时间步长$ρ_k$的轻度约束下，例如$ 0.4573328 \ leqρ_k\ leq 3.5615528 $ for $ k \ geq 2 $，与L2偏差的运算符相关的至关重要的双线性形式的积极半fientienties for。此结果使我们能够得出长时间的$ h^1 $ - l2方案的稳定性。这些积极的半足迹和$ h^1 $稳定性属性适用于标准分级网格，分级参数$ 1 <r \ leq 3.2016538 $。此外，还提供了$ h^1 $ norm中的一般非均匀网格的错误分析，并在$ h^1 $ -norm中的订单$（5-α）/2 $收敛时，当$ r> 5/α-1$时进行了修改的分级网格。据我们所知，这项研究是$ h^1 $ norm稳定性和L2方法的第一批作品，该方法是对亚扩散方程的一般不均匀网格的收敛。
This work establishes $H^1$-norm stability and convergence for an L2 method on general nonuniform meshes when applied to the subdiffusion equation. Under mild constraints on the time step ratio $ρ_k$, such as $0.4573328\leq ρ_k\leq 3.5615528$ for $k\geq 2$, the positive semidefiniteness of a crucial bilinear form associated with the L2 fractional-derivative operator is proved. This result enables us to derive long time $H^1$-stability of L2 schemes. These positive semidefiniteness and $H^1$-stability properties hold for standard graded meshes with grading parameter $1<r\leq 3.2016538$. In addition, error analysis in the $H^1$-norm for general nonuniform meshes is provided, and convergence of order $(5-α)/2$ in $H^1$-norm is proved for modified graded meshes when $r>5/α-1$. To the best of our knowledge, this study is the first work on $H^1$-norm stability and convergence of L2 methods on general nonuniform meshes for the subdiffusion equation.
