论文标题
向后扩散问题的数值分析
Numerical Analysis of Backward Subdiffusion Problems
论文作者
论文摘要
本文的目的是开发和分析数值方案，以近似于及时解决涉及分数导数的向后问题，该方程及时以$α\ in（0,1）$及时解决。在使用准界价值方法将“轻度”不适合问题的问题正常化后，我们通过在空间和卷积正交（CQ）中应用有限元法（FEM）提出了完全离散的方案。在两个平滑和非平滑数据的情况下，我们对所得离散系统进行了彻底的错误分析。该分析在很大程度上依赖于（离散）解决方案运算符的平滑属性，而在问题数据规则性方面，直接问题的非标准误差估计。理论结果对于平衡离散参数，正则化参数和噪声水平很有用。提出了数值示例以说明理论结果。
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 $α\in(0,1)$. 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.
