High-accuracy finite element method for 2D time fractional diffusion-wave equation on anisotropic meshes

References

• T. Apel, Anisotropic Finite Elements: Local Estimates and Applications, Stuttgart, Teubner, 1999.
   Google Scholar
• T. Apel and D. Sirch, L2-error estimates for Dirichlet and Neumann problems on anisotropic finite element meshes, Appl. Math. 56 (2011), pp. 177–206. doi: 10.1007/s10492-011-0002-7
   Google Scholar
• D.A. Benson, S.W. Wheatcraft, and M.M. Meerchaert, Application of a fractional advection-dispersion equation, Water Resour. Res. 36 (2000), pp. 1403–1412. doi: 10.1029/2000WR900031
   Web of Science ®Google Scholar
• A.H. Bhrawy, E.H. Doha, D. Baleanu, and S.S. Ezz-Eldien, A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations, J. Comput. Phys. 293 (2015), pp. 142–156. doi: 10.1016/j.jcp.2014.03.039
   Web of Science ®Google Scholar
• W. Bu, X. Liu, Y. Tang, and J. Yang, Finite element multigrid method for multi-term time fractional advection diffusion equations, Int. J. Model. Simul. Sci. Comput. 6 (2015), pp. 1540001–1540000. doi: 10.1142/S1793962315400012
   Web of Science ®Google Scholar
• W. Bu, Y. Tang, Y. Wu, and J. Yang, Finite difference/finite element methods for two-dimensional space and time fractional Bloch-Torrey equations, J. Comput. Phys. 293 (2015), pp. 264–279. doi: 10.1016/j.jcp.2014.06.031
   Web of Science ®Google Scholar
• H. Chen, S. L”u, and W. Chen, Spectral methods for the time fractional diffusion-wave equation in a semi-infinite channel, Comput. Math. Appl. 71 (2016), pp. 1818–1830. doi: 10.1016/j.camwa.2016.02.024
   Web of Science ®Google Scholar
• M. Dehghan, M. Abbaszadeh, and A. Mohebbi, Analysis of two methods based on Galerkin weak form for fractional diffusion-wave: Meshless interpolating element free Galerkin (IEFG) and finite element methods, Eng. Anal. Bound. Elem. 64 (2016), pp. 205–221. doi: 10.1016/j.enganabound.2015.11.011
   Web of Science ®Google Scholar
• J. Douglas and F. Milner, Interior and superconvergence estimates for mixed methods for second order elliptic equations, RAIRO Moder. Math. Anal. Numer. 19 (1985), pp. 297–328.
   Web of Science ®Google Scholar
• G. Fairweather, X. Yang, D. Xu, and H. Zhang, An ADI Crank–Nicolson orthogonal spline collocation method for the two-dimensional fractional diffusion-wave equation, J. Sci. Comput. 65 (2015), pp. 1217–1239. doi: 10.1007/s10915-015-0003-x
   Web of Science ®Google Scholar
• M.H. Heydari, M.R. Hooshmandasl, F.M. Maalek Ghaini, and C. Cattani, Wavelets method for the time fractional diffusion-wave equation, Phys. Lett. A 379 (2015), pp. 71–76. doi: 10.1016/j.physleta.2014.11.012
   Web of Science ®Google Scholar
• R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 1999.
   Google Scholar
• M.R. Hooshmandasl, M.H. Heydari, and C. Cattani, Numerical solution of fractional sub-diffusion and time-fractional diffusion-wave equations via fractional-order Legendre functions, Eur. Phys. J. Plus 131 (2016), pp. 268–260. (22 pages). doi: 10.1140/epjp/i2016-16268-2
   Web of Science ®Google Scholar
• J. Huang, Y. Tang, L. Vázquez, and J. Yang, Two finite difference schemes for time fractional diffusion-wave equation, Numer. Algor. 64 (2013), pp. 707–720. doi: 10.1007/s11075-012-9689-0
   Web of Science ®Google Scholar
• Y. Huang, J. Li, C. Wu, and W. Yang, Superconvergence analysis for linear tetrahedral edge elements, J. Sci. Comput. 62 (2015), pp. 122–145. doi: 10.1007/s10915-014-9848-7
   Web of Science ®Google Scholar
• W. Huang, L. Kamenski, and J. Lang, Stability of explicit one-step methods for P1-finite element approximation of linear diffusion equations on anisotropic meshes, SIAM J. Numer. Anal. 54 (2016), pp. 1612–1634. doi: 10.1137/130949531
   Google Scholar
• B. Jin, R. Lazarov, Y. Liu, and Z. Zhou, The Galerkin finite element method for a multi-term time-fractional diffusion equation, J. Comput. Phys. 281 (2015), pp. 825–843. doi: 10.1016/j.jcp.2014.10.051
   Web of Science ®Google Scholar
• A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
   Google Scholar
• N.N. Leonenko, M.M. Meerschaert, and A. Sikorskii, Fractional Pearson diffusions, J. Math. Anal. Appl. 403 (2013), pp. 532–546. doi: 10.1016/j.jmaa.2013.02.046
   PubMed Web of Science ®Google Scholar
• C. Li and F. Zeng, Numerical Methods for Fractional Calculus, CRC Press, Boca Raton, 2015.
   Google Scholar
• L. Li, D. Xu, and M. Luo, Alternating direction implicit Galerkin finite element method for the two-dimensional fractional diffusion-wave equation, J. Comput. Phys. 255 (2013), pp. 471–485. doi: 10.1016/j.jcp.2013.08.031
   Web of Science ®Google Scholar
• Q. Lin and J. Lin, Finite Element Methods: Accuracy and Improvement, Science Press, Beijing, 2006.
   Google Scholar
• Q. Lin and N. Yan, The Construction and Analysis of High Efficiency Finite Element Methods, Hebei University Press, Baoding, 1996.
   Google Scholar
• F. Liu, V. Anh, and I. Turner, Numerical solution of the space fractional Fokker–Planck Equation, J. Comput. Appl. Math. 166 (2004), pp. 209–219. doi: 10.1016/j.cam.2003.09.028
   Web of Science ®Google Scholar
• Q. Lin, L. Tobiska, and A. Zou, Superconvergence and extrapolation of non-conforming low order finite elements applied to the Poisson equation, IMA J. Numer. Anal. 25 (2005), pp. 160–181. doi: 10.1093/imanum/drh008
   Web of Science ®Google Scholar
• F. Liu, M.M. Meerschaert, R. McGough, P. Zhuang, and Q. Liu, Numerical methods for solving the multi-term time-fractional wave-diffusion equation, Fract. Calc. Appl. Anal. 16 (2013), pp. 9–25. doi: 10.2478/s13540-013-0002-2
   PubMed Web of Science ®Google Scholar
• F. Liu, P. Zhuang, and Q. Liu, Numerical Methods of Fractional Partial Differential Equations and Applications, Science Press, Beijing, 2015.
   Google Scholar
• M.M. Meerschaert, R.L. Magin, and A.Q Ye, Anisotropic fractional diffusion tensor imaging, J. Vib. Control. 22 (2016), pp. 2211–2221. doi: 10.1177/1077546314568696
   PubMed Web of Science ®Google Scholar
• K. Mustapha and W. Mclean, Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations, SIAM J. Numer. Anal. 51 (2013), pp. 491–515. doi: 10.1137/120880719
   Web of Science ®Google Scholar
• K. Mustapha, M. Nour, and B. Cockburn, Convergence and superconvergence analyses of HDG methods for time fractional diffusion problems, Adv. Copmut. Math.( 2015). doi: 10.1007/s10444-015-9428-x.
   Google Scholar
• S. Nemati and S. Sedaghat, Matrix method based on the second kind Chebyshev polynomials for solving time fractional diffusion-wave equations, J. Appl. Math. Comput. 51 (2016), pp. 189–207. doi: 10.1007/s12190-015-0899-1
   Web of Science ®Google Scholar
• V. Paolo and D. Boris, Plane anisotropic rari-constant materials, Math. Methods Appl. Sci. 39 (2016), pp. 3271–3281. doi: 10.1002/mma.3770
   Web of Science ®Google Scholar
• D. Shi and Y. Zhao, Quasi-Wilson nonconforming element approximation for nonlinear dual phase lagging heat conduction equations, Appl. Math. Comput. 243 (2014), pp. 454–464.
   Web of Science ®Google Scholar
• D. Shi, P. Wang, and Y. Zhao, Superconvergence analysis of anisotropic linear triangular finite element for nonlinear Schr”oinger equation, Appl. Math. Lett. 38 (2014), pp. 129–134. doi: 10.1016/j.aml.2014.07.019
   Web of Science ®Google Scholar
• F. Shojai, M. Kohandel, and A. Stepanian, On the relativistic anisotropic configurations, Eur. Phys. J. C 76 (2016), pp. 347–362. doi: 10.1140/epjc/s10052-016-4204-8
   Web of Science ®Google Scholar
• Z. Sun and X. Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math. 56 (2006), pp. 193–209. doi: 10.1016/j.apnum.2005.03.003
   Web of Science ®Google Scholar
• V.V. Uchaikin, Fractional Derivatives for Physicists and Engineers, Higher Education Press, Beijing, 2012.
   Google Scholar
• S. Vong and Z. Wang, A high order compact finite difference scheme for time fractional Fokker–Planck equations, Appl. Math. Lett. 43 (2015), pp. 38–43. doi: 10.1016/j.aml.2014.11.007
   Web of Science ®Google Scholar
• Z. Wang and S. Vong, Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation, J. Comput. Phys. 277 (2014), pp. 1–15. doi: 10.1016/j.jcp.2014.08.012
   Web of Science ®Google Scholar
• Z. Wang and S. Vong, A high-order ADI scheme for the two-dimensional time fractional diffusion-wave equation, Int. J. Comput. Math. 92 (2015), pp. 970–979. doi: 10.1080/00207160.2014.915960
   Web of Science ®Google Scholar
• J.Y. Yang, Y.M. Zhao, N. Liu, W.P. Bu, T.L. Xu, and Y.F. Tang, An implicit MLS meshless method for 2-D time dependent fractional diffusion-wave equation, Appl. Math. Model. 39 (2015), pp. 1229–1240. doi: 10.1016/j.apm.2014.08.005
   Web of Science ®Google Scholar
• C. Yao, J. Li, and D. Shi, Superconvergence analysis of nonconforming mixed finite element methods for time-dependent Maxwell's equations in isotropic cold plasma media, Appl. Math. Comput. 219 (2013), pp. 6466–6472.
   Web of Science ®Google Scholar
• W. Yang, D. Wang, and L. Yang, A stable numerical method for space fractional Landau–Lifshitz equations, Appl. Math. Lett. 61 (2016), pp. 149–155. doi: 10.1016/j.aml.2016.05.014
   Web of Science ®Google Scholar
• H. Ye, F. Liu, and V. Anh, Compact difference scheme for distributed-order time-fractional diffusion-wave equation on bounded domains, J. Comput. Phys. 298 (2015), pp. 652–660. doi: 10.1016/j.jcp.2015.06.025
   Web of Science ®Google Scholar
• F. Zeng, Second-order stable finite difference schemes for the time-fractional diffusion-wave equation, J. Sci. Comput. 65 (2015), pp. 411–430. doi: 10.1007/s10915-014-9966-2
   Web of Science ®Google Scholar
• Y. Zhang and D. Shi, Superconvergence of an H1-Galerkin nonconforming finite element method for a parabolic equation, Comput. Math. Appl. 66 (2013), pp. 2362–2375. doi: 10.1016/j.camwa.2013.09.013
   Google Scholar
• Y. Zhang, Z. Sun, and X. Zhao, Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation, SIAM J. Numer. Anal. 50 (2012), pp. 1535–1555. doi: 10.1137/110840959
   Web of Science ®Google Scholar
• Y. Zhao, W. Bu, J. Huang, D. Liu, and Y. Tang, Finite element method for two-dimensional space-fractional advection-dispersion equations, Appl. Math. Comput. 257 (2015), pp. 553–565.
   Web of Science ®Google Scholar
• Y. Zhao, P. Chen, W. Bu, X. Liu, and Y. Tang, Two mixed finite element methods for time-fractional diffusion equations, J. Sci. Comput. 70 (2017), pp. 407–428. doi: 10.1007/s10915-015-0152-y
   Web of Science ®Google Scholar
• Y. Zhao, Y. Zhang, F. Liu, I. Turner, Y. Tang, and V. Anh, Convergence and superconvergence of a fully-discrete scheme for multi-term time fractional diffusion equations, Comput. Math. Appl. 73 (2017), pp. 1087–1099. doi: 10.1016/j.camwa.2016.05.005
   Web of Science ®Google Scholar
• Y. Zhao, Y. Zhang, F. Liu, I. Turner, and D. Shi, Analytical solution and nonconforming finite element approximation for the 2D multi-term fractional subdiffusion equation, Appl. Math. Model. 40 (2016), pp. 8810–8825. doi: 10.1016/j.apm.2016.05.039
   Web of Science ®Google Scholar
• M. Zheng, F. Liu, V. Anh, and I. Turner, A high-order spectral method for the multi-term time-fractional diffusion equations, Appl. Math. Model. 40 (2016), pp. 4970–4985. doi: 10.1016/j.apm.2015.12.011
   Web of Science ®Google Scholar