Numerical analysis of backward subdiffusion problems. (24th September 2020)

• Record Type: Journal Article
• Authors: Zhang, Zhengqi; Zhou, Zhi
• Is Part Of: Inverse problems. Volume 36:Number 10(2020)
• Publication Date: 2020-09-24
• Subjects: fractional subdiffusion -- backward problem -- quasi-boundary value method -- finite element method -- convolution quadrature -- error analysis; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/abaf3d
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: 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 ∈ (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.