Galerkin finite element method for higher dimensional multi-term fractional diffusion equation on non-uniform meshes

References

• Ross B. Fractional calculus and its applications. 1st ed. Lecture notes in mathematics. Berlin: Springer Verlag; 1975.
   Google Scholar
• Oldham KB, Spanier J. The fractional calculus: theory and applications of differentiation and integration to arbitrary order. Dover books on mathematics. New York (NY): Dover; 2006.
   Google Scholar
• Mainardi F. Fractional calculus and waves in linear viscoelasticity. London: Imperial College Press; 2010.
   Google Scholar
• Podlubny I. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. 1st ed. Vol. 198, Mathematics in science and engineering. New York (NY): Academic Press; 1999.
   Google Scholar
• Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractional differential equations. 1st ed. Vol. 204, North-Holland mathematics studies. Boston: Elsevier; 2006.
   Google Scholar
• Chen C, Liu F, Burrage K. Finite difference methods and a Fourier analysis for the fractional reaction--subdiffusion equation. Appl. Math. Comput. 2008;198:754–769.
   Web of Science ®Google Scholar
• Meerschaert MM, Tadjeran C. Finite difference approximations for two-sided space-fractional partial differential equations. Appl. Numer. Math. 2006;56:80–90.
   Web of Science ®Google Scholar
• Lin R, Liu F, Anh V, et al. Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation. Appl. Math. Comput. 2009;212:435–445.
   Web of Science ®Google Scholar
• Zhuang P, Liu F, Anh V, et al. Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM J. Numer. Anal. 2009;47:1760–1781.
   Web of Science ®Google Scholar
• Lin Y, Xu C. Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 2007;225:1533–1552.
   Web of Science ®Google Scholar
• Li X, Xu C. A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 2009;47:2108–2131.
   Web of Science ®Google Scholar
• Yang X, Zhang H, Xu D. Orthogonal spline collocation method for the two-dimensional fractional sub-diffusion equation. J. Comput. Phys. 2014;256:824–837.
   Web of Science ®Google Scholar
• Zhang H, Liu F, Anh V. Galerkin finite element approximation of symmetric space-fractional partial differential equations. Appl. Math. Comput. 2010;217:2534–2545.
   Web of Science ®Google Scholar
• Zhao Z, Li C. Fractional difference/finite element approximations for the time-space fractional telegraph equation. Appl. Math. Comput. 2012;219:2975–2988.
   Web of Science ®Google Scholar
• Deng W. Finite element method for the space and time fractional Fokker--Planck equation. SIAM J. Numer. Anal. 2008;47:204–226.
   Web of Science ®Google Scholar
• Jiang Y, Ma J. High-order finite element methods for time-fractional partial differential equations. J. Comput. Appl. Math. 2011;235:3285–3290.
   Web of Science ®Google Scholar
• Jiang Y, Ma J. Moving finite element methods for time fractional partial differential equations. Sci. China Math. 2013;56:1287–1300.
   Web of Science ®Google Scholar
• Badr AA. Finite element method for linear multiterm fractional differential equations. J. Appl. Math. 2012.
   Web of Science ®Google Scholar
• Liu F, Meerschaert MM, McGough RJ, et al. Numerical methods for solving the multi-term time-fractional wave-diffusion equation. Fract. Calc. Appl. Anal. 2013;16:9–25.
   PubMed Web of Science ®Google Scholar
• Jin B, Lazarov R, Liu Y, et al. The Galerkin finite element method for a multi-term time-fractional diffusion equation. J. Comput. Phys. 2015;281:825–843.
   Web of Science ®Google Scholar
• Bu W, Liu X, Tang Y, et al. Finite element multigrid method for multi-term time fractional advection diffusion equations. Int. J. Mod. Sim. Sci. Comp. 2015;6:1540001.
   Web of Science ®Google Scholar
• Zhang Y, Sun Z. Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation. J. Comput. Phys. 2011;230:8713–8728.
   Web of Science ®Google Scholar
• Zhang Y, Sun Z, Zhao X. Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation. SIAM J. Numer. Anal. 2012;50:1535–1555.
   Web of Science ®Google Scholar
• Zhang N, Deng W, Wu Y. Finite difference/element method for a two-dimensional modified fractional diffusion equation. Adv. Appl. Math. Mech. 2012;4:496–518.
   Web of Science ®Google Scholar
• Bu W, Tang Y, Wu Y, et al. Finite difference/finite element method for two-dimensional space and time fractional Bloch--Torrey equations. J. Comp. Phys. 2015;293:264–279.
   Web of Science ®Google Scholar
• Zhang Y, Sun Z, Liao H. Finite difference methods for the time fractional diffusion equation on non-uniform meshes. J. Comput. Phys. 2014;265:195–210.
   Web of Science ®Google Scholar
• Thomée V. Galerkin finite element methods for parabolic problems. 2nd ed. Vol. 25, Springer series in computational mathematics. Berlin: Springer; 1997.
   Google Scholar