ANISOTROPIC EQ1ROT FINITE ELEMENT APPROXIMATION FOR A MULTI-TERM TIME-FRACTIONAL MIXED SUB-DIFFUSION AND DIFFUSION-WAVE EQUATION

doi:10.4208/jcm.2110-m2021-0180

Journal of Computational Mathematics ›› 2023, Vol. 41 ›› Issue (3): 459-482.DOI: 10.4208/jcm.2110-m2021-0180

ANISOTROPIC EQ₁^ROT FINITE ELEMENT APPROXIMATION FOR A MULTI-TERM TIME-FRACTIONAL MIXED SUB-DIFFUSION AND DIFFUSION-WAVE EQUATION

Huijun Fan^1,2, Yanmin Zhao^1,3, Fenling Wang^1,3, Yanhua Shi^1,3, Fawang Liu^4,5

1. School of Science, Xuchang University, Xuchang 461000, China;
2. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China;
3. Henan Joint International Research Laboratory of High Performance Computation for Complex Systems, Xuchang 461000, China;
4. School of Mathematical Sciences, Queensland University of Technology, Brisbane QLD 4001, Australia;
5. School of Mathematics and Statistics, Fuzhou University, Fuzhou 350108, China

Received:2021-06-22 Revised:2021-09-07 Published:2023-03-14
Contact: Yanmin Zhao, Email:zhaoym@lsec.cc.ac.cn
Supported by:
The work is supported by the National Natural Science Foundation of China (No. 11971416), the Scientific Research Innovation Team of Xuchang University (No. 2022CXTD002), the Foundation for University Key Young Teacher of Henan Province (No. 2019GGJS214), the Key Scientific Research Projects in Universities of Henan Province (Nos. 21B110007, 22A110022), the National Natural Science Foundation of China (International cooperation key project:No. 12120101001) and the Australian Research Council via the Discovery Project (DP190101889).

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Huijun Fan, Yanmin Zhao, Fenling Wang, Yanhua Shi, Fawang Liu. ANISOTROPIC EQ₁^ROT FINITE ELEMENT APPROXIMATION FOR A MULTI-TERM TIME-FRACTIONAL MIXED SUB-DIFFUSION AND DIFFUSION-WAVE EQUATION[J]. Journal of Computational Mathematics, 2023, 41(3): 459-482.

By employing EQ₁^rot nonconforming finite element, the numerical approximation is presented for multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on anisotropic meshes. Comparing with the multi-term time-fractional sub-diffusion equation or diffusion-wave equation, the mixed case contains a special time-space coupled derivative, which leads to many difficulties in numerical analysis. Firstly, a fully discrete scheme is established by using nonconforming finite element method (FEM) in spatial direction and L1 approximation coupled with Crank-Nicolson (L1-CN) scheme in temporal direction. Furthermore, the fully discrete scheme is proved to be unconditional stable. Besides, convergence and superclose results are derived by using the properties of EQ₁^rot nonconforming finite element. What's more, the global superconvergence is obtained via the interpolation postprocessing technique. Finally, several numerical results are provided to demonstrate the theoretical analysis on anisotropic meshes.

Key words: Multi-term time-fractional mixed sub-diffusion and diffusion-wave equation, Nonconforming FEM, L1-CN scheme, Anisotropic meshes, Convergence and superconvergence

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ANISOTROPIC EQ1ROT FINITE ELEMENT APPROXIMATION FOR A MULTI-TERM TIME-FRACTIONAL MIXED SUB-DIFFUSION AND DIFFUSION-WAVE EQUATION

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ANISOTROPIC EQ₁^ROT FINITE ELEMENT APPROXIMATION FOR A MULTI-TERM TIME-FRACTIONAL MIXED SUB-DIFFUSION AND DIFFUSION-WAVE EQUATION