A two-grid virtual element method for nonlinear variable-order time-fractional diffusion equation on polygonal meshes

References

• L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L.D. Marini, and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23(1) (2013), pp. 199–214.
   Web of Science ®Google Scholar
• L. Beirão da Veiga, F. Brezzi, L.D. Marini, and A. Russo, The Hitchhiker's guide to the virtual element method, Math. Models Methods Appl. Sci. 24(08) (2014), pp. 1541–1573.
   Google Scholar
• L. Beirão da Veiga, F. Dassi, and G. Vacca, The stokes complex for virtual elements in three dimensions, Math. Models Methods Appl. Sci. 30(03) (2020), pp. 477–512.
   Google Scholar
• L. Beirão da Veiga, F. Dassi, G. Manzini, and L. Mascotto, Virtual elements for Maxwell's equations, Comput. Math. Appl. 116 (2022), pp. 82–99.
   Web of Science ®Google Scholar
• C. Bi and V. Ginting, Two-grid finite volume element method for linear and nonlinear elliptic problems, Numer. Math. 108(2) (2007), pp. 177–198.
   Web of Science ®Google Scholar
• L. Chen and J.G. Huang, Some error analysis on virtual element methods, Calcolo 55(1) (2018), pp. 1–23.
   Web of Science ®Google Scholar
• Y. Chen, Y. Huang, and D. Yu, A two-grid method for expanded mixed finite-element solution of semilinear reaction–diffusion equations, Int. J. Numer. Methods Eng. 57(2) (2003), pp. 193–209.
   Web of Science ®Google Scholar
• Y. Chen, P. Luan, and Z. Lu, Analysis of two-grid methods for nonlinear parabolic equations by expanded mixed finite element methods, Adv. Appl. Math. Mech. 1 (2009), pp. 830–844.
   Web of Science ®Google Scholar
• F.X. Chen, Q.M. Wang, and Z.J. Zhou, Two-grid virtual element discretization of semilinear elliptic problem, Appl. Numer. Math. 186 (2023), pp. 228–240.
   Web of Science ®Google Scholar
• Q. Gu, Y. Chen, and Y. Huang, Superconvergence analysis of a two-grid finite element method for nonlinear time fractional diffusion equations, Comput. Appl. Math. 41(8) (2022), pp. 361.
   Google Scholar
• C. Huang and H. Chen, Superconvergence analysis of finite element methods for the variable-order subdiffusion equation with weakly singular solutions, Appl. Math. Lett. 139 (2023), pp. 108559.
   Web of Science ®Google Scholar
• T. Lee, L. Bocquet, and B. Coasne, Activated desorption at heterogeneous interfaces and long-time kinetics of hydrocarbon recovery from nanoporous media, Nat. Commun. 7(1) (2016), pp. 11890.
   PubMedGoogle Scholar
• Q. Li, Y. Chen, Y. Huang, and Y. Wang, Two-grid methods for semilinear time fractional reaction diffusion equations by expanded mixed finite element method, Appl. Numer. Math. 157 (2020), pp. 38–54.
   Web of Science ®Google Scholar
• Meng Li, J.K. Zhao, C.M. Huang, and S.C. Chen, Conforming and nonconforming VEMs for the fourth-order reaction-subdiffusion equation: A unified framework, IMA J. Numer. Anal. 42(3) (2022), pp. 2238–2300.
   Web of Science ®Google Scholar
• Q. Li, Y. Chen, Y. Huang, and Y. Wang, Two-grid methods for nonlinear time fractional diffusion equations by L1-Galerkin FEM, Math. Comput. Simul. 185 (2021), pp. 436–451.
   Web of Science ®Google Scholar
• Y. Liu, Y. Du, H. Li, J. Li, and S. He, A two-grid mixed finite element method for a nonlinear fourth-order reaction-diffusion problem with time-fractional derivative, Comput. Math. Appl. 70(10) (2015), pp. 2474–2492.
   Web of Science ®Google Scholar
• Y. Liu, Y. Du, H. Li, and J. Wang, A two-grid finite element approximation for a nonlinear time- fractional cable equation, Nonlinear Dyn. 85(4) (2016), pp. 2535–2548.
   Web of Science ®Google Scholar
• H. Liu, A. Cheng, and H. Wang, A parareal finite volume method for variable-order time-fractional diffusion equations, J. Sci. Comput. 85(1) (2020), pp. 19.
   Web of Science ®Google Scholar
• C. Lorenzo and T. Hartley, Variable order and distributed order fractional operators, Nonlinear Dyn. 29(1/4) (2002), pp. 57–98.
   Web of Science ®Google Scholar
• H. Sun, W. Chen, and Y. Chen, Variable-order fractional differential operators in anomalous diffusion modeling, Phys. A Stat. Mech. Appl. 388(21) (2009), pp. 4586–4592.
   Web of Science ®Google Scholar
• G. Vacca and L. Beirão da Veiga, Virtual element methods for parabolic problems on polygonal meshes, Numer. Methods Partial Differ. Equ. 31(6) (2015), pp. 2110–2134.
   Web of Science ®Google Scholar
• H. Wang and X. Zheng, Analysis and numerical solution of a nonlinear variable-order fractional differential equation, Adv. Comput. Math. 45(5-6) (2019), pp. 2647–2675.
   Web of Science ®Google Scholar
• H. Wang and X. Zheng, Well-posedness and regularity of the variable-order time-fractional diffusion equations, J. Math. Anal. Appl. 475(2) (2019), pp. 1778–1802.
   Web of Science ®Google Scholar
• Y. Wei, Y. Zhao, H. Chen, F. Wang, and S. Lu, On the convergence and superconvergence for a class of two-dimensional time fractional reaction-subdiffusion equations, Numer. Methods Partial Differ. Equ. 39(1) (2023), pp. 481–500.
   Web of Science ®Google Scholar
• J. Xu, A novel two-grid method for semilinear equations, SIAM J. Sci. Comput. 15 (1994), pp. 231–237.
   Web of Science ®Google Scholar
• J. Xu, Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal. 33 (5) (1996), pp. 1759–1777.
   Web of Science ®Google Scholar
• X. Yang, L. Wu, and H. Zhang, A space-time spectral order sinc-collocation method for the fourth-order nonlocal heat model arising in viscoelasticity, Appl. Math. Comput. 457 (2023), pp. 128192.
   Web of Science ®Google Scholar
• F. Zeng, Z. Zhang, and G.K. Arniadakis, A generalized spectral collocation method with tunable accuracy for variable-order fractional differential equations, SIAM J. Sci. Comput. 37(6) (2015), pp. A2710–A2732.
   Web of Science ®Google Scholar
• B. Zhang and M. Feng, Virtual element method for two-dimensional linear elasticity problem in mixed weakly symmetric formulation, Appl. Math. Comput. 328 (2018), pp. 1–25.
   Web of Science ®Google Scholar
• Y.D. Zhang and M.F. Feng, A mixed virtual element method for the time-fractional fourth-order subdiffusion equation, Numer. Algorithms (90)  (2022), pp. 1617–1637.
   Google Scholar
• Y.D. Zhang and M.F. Feng, A local projection stabilization virtual element method for the time-fractional burgers equation with high reynolds numbers, Appl. Math. Comput. 436 (2023), pp. 127509.
   Web of Science ®Google Scholar
• X. Zheng and H. Wang, Optimal-order error estimates of finite element approximations to variable-order time-fractional diffusion equations without regularity assumptions of the true solutions, IMA J. Numer. Anal. 41(2) (2021), pp. 1522–1545.
   Web of Science ®Google Scholar
• P. Zhuang, F. Liu, V. Anh, and I. Turner, Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term, SIAM J. Numer. Anal. 47(3) (2009), pp. 1760–1781.
   Web of Science ®Google Scholar