The aim of this paper is to develop and analyze numerical schemes for approximately solving the backward problem of subdiffusion equation involving a fractional derivative in time with order $\alpha\in(0,1)$. After using quasi-boundary value method to regularize the "mildly" ill-posed problem, we propose a fully discrete scheme by applying finite element method (FEM) in space and convolution quadrature (CQ) in time. We provide a thorough error analysis of the resulting discrete system in both cases of smooth and nonsmooth data. The analysis relies heavily on smoothing properties of (discrete) solution operators, and nonstandard error estimate for the direct problem in terms of problem data regularity. The theoretical results are useful to balance discretization parameters, regularization parameter and noise level. Numerical examples are presented to illustrate the theoretical results.

中文翻译：

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后向亚扩散问题的数值分析

本文的目的是开发和分析数值方案，以近似求解包含阶为 $\alpha\in(0,1)$ 的时间分数阶导数的子扩散方程的后向问题。在使用准边界值方法对“轻度”不适定问题进行正则化后，我们通过在空间上应用有限元方法（FEM）和在时间上应用卷积正交（CQ），提出了一个完全离散的方案。我们在光滑和非光滑数据的两种情况下都提供了对所得离散系统的彻底误差分析。该分析在很大程度上依赖于（离散）解算子的平滑特性，以及在问题数据规律性方面对直接问题的非标准误差估计。理论结果有助于平衡离散化参数、正则化参数和噪声水平。