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

Huijun Fan1,2, Yanmin Zhao1,3, Fenling Wang1,3, Yanhua Shi1,3, Fawang Liu4,5

1. 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).

Huijun Fan, Yanmin Zhao, Fenling Wang, Yanhua Shi, Fawang Liu. ANISOTROPIC EQ1ROT 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 EQ1rot 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 EQ1rot 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.

CLC Number:

 [1] Y. Liu, Z. Yu, H. Li, F. Liu, J. Wang, Time two-mesh algorithm combined with finite element method for time fractional water wave model, Int. J. Heat. Mass. Tran., 120(2018), 1132-1145.[2] S. Qin, F. Liu, I. Turner, V. Vegh, Q. Yu, Q. Yang, Multi-term time-fractional Bloch equations and application in magnetic resonance imaging, J. Comput. Appl. Math., 319(2017), 308-319.[3] H. Wang, C. Weng, Z. Song, J. Cai, Research on the law of spatial fractional calculus diffusion equation in the evolution of chaotic economic system, Chaos Solitons Fractals, 131(2020), 109462.[4] Tomasz P. Stefański, J. Gulgowski, Signal propagation in electromagnetic media described by fractional-order models, Commun. Nonlinear Sci. Numer. Simul., 82(2020), 105029.[5] A. Zhokh, P. Strizhak, Macroscale modeling the methanol anomalous transport in the porous pellet using the time-fractional diffusion and fractional Brownian motion:A model comparison, Commun. Nonlinear Sci. Numer. Simul., 79(2019), 104922.[6] L. Li, D. Li, Exact solutions and numerical study of time fractional Burgers' equations, Appl. Math. Lett., 100(2020), 106011.[7] A. Jannelli, M. Ruggieri, M. P. Speciale, Analytical and numerical solutions of time and space fractional advection-diffusion-reaction equation, Commun. Nonlinear Sci. Numer. Simul., 70(2019), 89-101.[8] W. Malesza, M. Macias, D. Sierociuk, Analytical solution of fractional variable order differential equations, J. Comput. Appl. Math., 348(2019), 214-236.[9] J. Zhang, F. Liu, Z. Lin, V. Anh, Analytical and numerical solutions of a multi-term time-fractional Burgers' fluid model, Appl. Math. Comput., 356(2019), 1-12.[10] X. Ding, Y. Jiang, Analytical solutions for multi-term time-space coupling fractional delay partial differential equations with mixed boundary conditions, Commun. Nonlinear Sci. Numer. Simul., 65(2018), 231-247.[11] M. Hussain, S. Haq, Weighted meshless spectral method for the solutions of multi-term time fractional advection-diffusion problems arising in heat and mass transfer, Int. J. Heat. Mass. Tran., 129(2019), 1305-1316.[12] R. Zheng, F. Liu, X. Jiang, A legendre spectral method on graded meshes for the two-dimensional multi-term time-fractional diffusion equation with non-smooth solutions, Appl. Math. Lett., 104(2020), 106247.[13] Y. Zhao, Y. Zhang, F. Liu, I. Turner, Y. Tang, V. Anh, Convergence and superconvergence of a fully-discrete scheme for multi-term time fractional diffusion equations, Comput. Math. Appl., 73(2017), 1087-1099.[14] B. Jin, R. Lazarov, Y. Liu, Z. Zhou, The Galerkin finite element method for a multi-term timefractional diffusion equation, J. Comput. Phys., 281(2015), 825-843.[15] L. Qiao, D. Xu, Orthogonal spline collocation scheme for the multi-term time-fractional diffusion equations, Int. J. Comput. Math., 95(2018), 1478-1493.[16] Y. Wang, X. Wen, A compact exponential difference method for multi-term time-fractional convection-reaction-diffusion problems with non-smooth solutions, Appl. Math. Comput., 381(2020), 125316.[17] L. Wei, Stability and convergence of a fully discrete local discontinuous Galerkin method for multi-term time fractional diffusion equations, Numer. Algorithms, 76(2017), 695-707.[18] A.S.V.R. Kanth, N. Garg, An implicit numerical scheme for a class of multi-term time-fractional diffusion equation, Eur. Phys. J. Plus., 134(2019). http://dx.doi.org/10.1140/epjp/i2019-12696-8.[19] L. Zhao, F. Liu, V. Anh, Numerical methods for the two-dimensional multi-term time-fractional diffusion equations, Comput. Math. Appl., 74(2017), 2253-2268.[20] Z. Fu, L. Yang, H. Zhu, W. Xu, A semi-analytical collocation Trefftz scheme for solving multi-term time fractional diffusion-wave equations, Eng. Anal. Bound. Elem., 98(2019), 137-146.[21] H. Sun, X. Zhao, Z. Sun, The temporal second order difference schemes based on the interpolation approximation for the time multi-term fractional wave equation, J. Sci. Comput., 78(2019), 467- 498.[22] H. Liu, S. Lü, Gauss-Lobatto-Legendre-Birkhoff pseudospectral approximations for the multiterm time fractional diffusion-wave equation with Neumann boundary conditions, Numer. Methods Partial Differential Equations, 34(2018), 2217-2236.[23] M.H. Heydari, Z. Avazzadeh, M. F. Haromi, A wavelet approach for solving multi-term variableorder time fractional diffusion-wave equation, Appl. Math. Comput., 341(2019), 215-228.[24] P. Lyu, Y. Liang, Z. Wang, A fast linearized finite difference method for the nonlinear multi-term time-fractional wave equation, Appl. Numer. Math., 151(2020), 448-471.[25] Z. Shi, Y. Zhao, F. Liu, F. Wang, Y. Tang, Nonconforming quasi-Wilson finite element method for 2D multi-term time fractional diffusion-wave equation on regular and anisotropic meshes, Appl. Math. Comput., 338(2018), 290-304.[26] Y. Zhao, F. Wang, X. Hu, Z. Shi, Y. Tang, Anisotropic linear triangle finite element approximation for multi-term time-fractional mixed diffusion and diffusion-wave equations with variable coefficient on 2D bounded domain, Comput. Math. Appl., 78(2019), 1705-1719.[27] S. Shen, F. Liu, V. Anh, The analytical solution and numerical solutions for a two-dimensional multi-term time fractional diffusion and diffusion-wave equation, J. Comput. Appl. Math., 345(2019), 515-534.[28] Z. Liu, F. Liu, F. Zeng, An alternating direction implicit spectral method for solving two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations, Appl. Numer. Math., 136(2019), 139-151.[29] Y. Liu, H. Sun, X. Yin, L. Feng, Fully discrete spectral method for solving a novel multi-term time-fractional mixed diffusion and diffusion-wave equation, Z. Angew. Math. Phys., 71(2020), http://dx.doi.org/10.1007/s00033-019-1244-6.[30] H. Chen, S. Lü, W. Chen, A unified numerical scheme for the multi-term time fractional diffusion and diffusion-wave equations with variable coefficients, J. Comput. Appl. Math., 330(2018), 380- 397.[31] Y. Zhang, Y. Zhao, F. Wang, Y. Tang, High-accuracy finite element method for 2D time fractional diffusion-wave equation on anisotropic meshes, Int. J. Comput. Math., 95(2018), 218-230.[32] F. Liu, P. Zhuang, Q. Liu, Numerical Methods of Fractional Partial Differential Equations and Applications, Science Press, Beijing, 2015.[33] C. Li, L. Zheng, Y. Zhang, L. Ma, X. Zhang, Helical flows of a heated generalized Oldroyd-B fluid subject to a time-dependent shear stress in porous medium, Commun. Nonlinear Sci. Numer. Simluat., 17(2012), 5026-5041.[34] M.B. Riaz, M.A. Imran, K. Shabbir, Analytic solutions of Oldroyd-B fluid with fractional derivatives in a circular duct that applies a constant couple, Alex. Eng. J., 55(2016), 3267-3275.[35] L. Feng, F. Liu, I. Turner, L. Zheng, Novel numerical analysis of multi-term time fractional viscoelastic non-Newtonian fluid models for simulating unsteady MHD Couette flow of a generalized Oldroyd-B fluid, Fract. Calc. Appl. Anal., 21(2018), 1073-1103.[36] Y. Liu, X. Yin, L. Feng, H. Sun, Finite difference scheme for simulating a generalized twodimensional multi-term time fractional non-Newtonian fluid model, Adv. Difference Equ., 2018(2018), 1-16.[37] Y. Zhang, J. Jiang, Y. Bai, MHD flow and heat transfer analysis of fractional Oldroyd-B nanofluid between two coaxial cylinders, Comput. Math. Appl., 78(2019), 3408-3421.[38] L. Feng, F. Liu, I. Turner, Finite difference/finite element method for a novel 2D multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on convex domains, Commun. Nonlinear Sci. Numer. Simul., 70(2019), 354-371.[39] H. Zhang, X. Yang, Superconvergence analysis of nonconforming finite element method for timefractional nonlinear parabolic equations on anisotropic meshes, Comput. Math. Appl., 77(2019), 2707-2724.[40] W. Huang, L. Kamenski, J. Lang, Conditioning of implicit Runge-Kutta integration for finite element approximation of linear diffusion equations on anisotropic meshes, J. Comput. Appl. Math., 387(2019), 112497.[41] Y. Wei, S. Lü, H. Chen, Y. Zhao, F. Wang, Convergence analysis of the anisotropic FEM for 2D time fractional variable coefficient diffusion equations on graded meshes, Appl. Math. Lett., 111(2021), 106604.[42] J. Wang, Superconvergence analysis of an energy stable scheme for nonlinear reaction-diffusion equation with BDF mixed FEM, Appl. Numer. Math., 153(2020), 457-472.[43] M. Li, D. Shi, L. Pei, Convergence and superconvergence analysis of finite element methods for the time fractional diffusion equation, Appl. Numer. Math., 151(2020), 141-160.[44] Y. Du, H. Wu, Z. Zhang, Superconvergence analysis of linear FEM based on polynomial preserving recovery for Helmholtz equation with high wave number, J. Comput. Appl. Math., 372(2020), 112731.[45] Q. Lin, L. Tobiska, A. Zhou, Superconvergence and extrapolation of non-conforming low order finite elements applied to the Possion equation, IMA J. Numer. Anal., 25(2005), 160-181.[46] D. Shi, S. Mao, S. Chen, An anisotropic nonconforming finite element with some superconvergence results, J. Comput. Math., 23(2005), 261-274.[47] Z. Sun, The Method of Order Reduction and its Application to the Numerical Solutions of Partial Differential Equations, Science Press, Beijing, 2009.[48] Z. Sun, X. Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 56(2006), 193-209.[49] D. Shi, Y. Zhang, High accuracy analysis of a new nonconforming mixed finite element scheme for Sobolev equations, Appl. Math. Comput., 218(2011), 3176-3186.
 [1] Yanping Chen, Qiling Gu, Qingfeng Li, Yunqing Huang. A TWO-GRID FINITE ELEMENT APPROXIMATION FOR NONLINEAR TIME FRACTIONAL TWO-TERM MIXED SUB-DIFFUSION AND DIFFUSION WAVE EQUATIONS [J]. Journal of Computational Mathematics, 2022, 40(6): 936-954. [2] Dongyang Shi, Chao Xu. AN ANISOTROPIC LOCKING-FREE NONCONFORMING TRIANGULAR FINITE ELEMENT METHOD FOR PLANAR LINEAR ELASTICITY PROBLEM [J]. Journal of Computational Mathematics, 2012, 30(2): 124-138. [3] Qingshan Li, Huixia Sun, Shaochun Chen . Convergence of a Mixed Finite Element for the Stokes Problem onAnisotropic Meshes [J]. Journal of Computational Mathematics, 2008, 26(5): 740-755. [4] Shi-peng Mao,Shao-chun Chen. CONVERGENCE ANALYSIS OF MORLEY ELEMENT ON ANISOTROPIC MESHES [J]. Journal of Computational Mathematics, 2006, 24(2): 169-180. [5] Dong-yang Shi, Shi-peng Mao, Shao-chun Chen . An Anisotropic Nonconforming Finite Element with Some Superconvergence Results [J]. Journal of Computational Mathematics, 2005, 23(3): 261-.
Viewed
Full text

Abstract