Superconvergence Analysis of Anisotropic FEMs for Time Fractional Variable Coefficient Diffusion Equations
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Superconvergence Analysis of Anisotropic FEMs for Time Fractional Variable Coefficient Diffusion Equations Yabing Wei1,2 · Yanmin Zhao3 · Fenling Wang3 · Yifa Tang4,5 · Jiye Yang6 Received: 20 November 2019 / Revised: 26 February 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract In this paper, based on Alikhanov’s L2-1σ high-order approximation and anisotropic finite element methods, a fully discrete scheme for time fractional variable coefficient diffusion equations on anisotropic meshes is presented. Firstly, we prove that the discrete scheme is unconditionally stable in H 1 -norm, then the results of convergence in L 2 -norm and superclose in H 1 -norm are derived by combining interpolation with projection, and then, the superconvergence in H 1 -norm is obtained by using interpolation post-processing technique. In addition, it is worth mentioning the key technology of combining interpolation and projection. If interpolation or projection is used alone, the results of this article cannot be obtained. Finally, numerical examples are provided to verify the correctness of theoretical analysis. Keywords L2-1σ approximation · Anisotropic meshes · Time fractional variable coefficient diffusion equations · Convergence and superconvergence Mathematics Subject Classification 65M60 · 65M12 · 35R11
Communicated by Theodore E. Simos.
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Yanmin Zhao [email protected]
1
School of Mathematical Sciences, Beihang University, Beijing 100083, China
2
State Key Laboratory of Space Weather, Chinese Academy of Sciences, Beijing 100190, China
3
School of Mathematics and Statistics, Xuchang University, Xuchang 461000, China
4
LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
5
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
6
School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China
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Y. Wei et al.
1 Introduction Fractional partial differential equations are equations containing non-integer derivatives and have a wide range of applications in real life. For example, fractional partial differential equations play key roles in image processing, such as image denoising (see [1,2]), image enhancement (see [3]) and image super-resolution reconstruction (see [4,5]). Especially in terms of denoising and reducing the staircase effect, the fractional derivative is more accurate than the integer one, which is also an irreplaceable advantage of the fractional partial differential equations. Moreover, studies have shown that fractional partial differential equations are very suitable for describing materials with memory and genetic properties such as dispersion in fractal and porous media, capacitance theory, electrolytic chemistry, semiconductor physics, viscoelastic systems, biomathematics and statistical mechanics (see [6–15]). With the increase in application fields of fractional partial differential equations, the study of fractional partial
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