Space-Fractional Diffusion Equation with Variable Coefficients: Well-posedness and Fourier Pseudospectral Approximation

Multi-dimensional space-fractional diffusion equation with variable coefficients and fractional gradient is a difficult problem in theory and computation. As far as we know, there rarely exist well-posedness results and efficient numerical approaches for such equation. In this paper, we focus on thi...

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Bibliographic Details
Published inJournal of scientific computing Vol. 87; no. 1
Main Authors Li, Xue-Yang, Xiao, Ai-Guo
Format Journal Article
LanguageEnglish
Published New York Springer US 01.04.2021
Subjects
Algorithms
Computational Mathematics and Numerical Analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Theoretical
Space-fractional diffusion equation
65T40
Variable coefficients
65M70
35R11
Fourier pseudospectral method
65M12
Well-posedness
Convergence
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Summary:Multi-dimensional space-fractional diffusion equation with variable coefficients and fractional gradient is a difficult problem in theory and computation. As far as we know, there rarely exist well-posedness results and efficient numerical approaches for such equation. In this paper, we focus on this subject. First, we apply the commutator estimation method to prove the coercivity of the non-positive bilinear form for such equation in both continuous sense and discrete sense, and this is key for the later discussion. Then, we prove the well-posedness of the analytical solution and give the global error estimation of the numerical solution obtained by Crank–Nicolson Fourier pseudospectral scheme. Last, the numerical experiments are used to verify the main results of the theoretical analysis, and a model for the plume of solute through groundwater is exhibited to show the application of space-fractional diffusion theory.
ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-021-01435-w