In this work, we develop an efficient incomplete iterative scheme for the numerical solution of the subdiffusion model involving a Caputo derivative of order $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) in time. It is based on piecewise linear Galerkin finite element method in space and backward Euler convolution quadrature in time and solves one linear algebraic system inexactly by an iterative algorithm at each time step. We present theoretical results for both smooth and nonsmooth solutions, using novel weighted estimates of the time-stepping scheme. The analysis indicates that with the number of iterations at each time level chosen properly, the error estimates are nearly identical with that for the exact linear solver, and the theoretical findings provide guidelines on the choice. Illustrative numerical results are presented to complement the theoretical analysis.

中文翻译：

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亚扩散的不完全迭代解

在这项工作中，我们为子扩散模型的数值解开发了一种有效的不完全迭代方案，该模型涉及时间阶 $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) 的 Caputo 导数。它基于空间上的分段线性伽辽金有限元法和时间上的后向欧拉卷积求积，通过迭代算法在每个时间步不精确地求解一个线性代数系统。我们使用时间步进方案的新加权估计来呈现平滑和非平滑解决方案的理论结果。分析表明，通过正确选择每个时间级别的迭代次数，误差估计与精确线性求解器的误差估计几乎相同，理论研究结果为选择提供了指导。