# zbMATH — the first resource for mathematics

High-order finite element methods for time-fractional partial differential equations. (English) Zbl 1216.65130
Summary: The aim of this paper is to develop high-order methods for solving time-fractional partial differential equations. The proposed high-order method is based on a high-order finite element method for space and a finite difference method for time. An optimal convergence rate of $$O((\Delta t)^{2 - \alpha }+N^{ - r})$$ is proved for the $$(r - 1)$$th-order finite element method ($$r\geq 2$$).

##### MSC:
 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35K05 Heat equation 35R11 Fractional partial differential equations 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
Full Text:
##### References:
 [1] Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010 [2] Jumarie, G., A fokker – planck equation of fractional order with respect to time, J. math. phys., 33, 3536-3542, (1992) · Zbl 0761.60071 [3] Gorenflo, R.; Mainardi, F.; Moretti, D.; Paradisi, P., Time fractional diffusion: a discrete random walk approach, Nonlinear dynam., 29, 129-143, (2002) · Zbl 1009.82016 [4] Diethelm, K.; Ford, N.J., Numerical solution of the bagley-torvik equation, Bit, 42, 490-507, (2002) · Zbl 1035.65067 [5] Zhuang, P.; Liu, F.; Anh, V.; Turner, I., New solution and analytical techniques of the implicit numerical methods for the anomalous sub-diffusion equation, SIAM J. numer. anal., 46, 1079-1095, (2008) · Zbl 1173.26006 [6] Liu, F.; Yang, C.; Burrage, K., Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term, J. comput. appl. math., 231, 160-176, (2009) · Zbl 1170.65107 [7] Liu, F.; Anh, V.; Turner, I.; Zhuang, P., Time fractional advection dispersion equation, J. comput. appl. math., 13, 233-245, (2003) · Zbl 1068.26006 [8] Schneider, W.R.; Wyss, W., Fractional diffusion and wave equations, J. math. phys., 30, 134-144, (1989) · Zbl 0692.45004 [9] Wyss, W., The fractional diffusion equation, J. math. phys., 27, 2782-2785, (1986) · Zbl 0632.35031 [10] Liu, F.; Shen, S.; Anh, V.; Turner, I., Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation, Anziam j., 46, E, 488-504, (2005) · Zbl 1082.60511 [11] Scherer, R.; Kalla, S.L.; Boyadjev, L.; Al-Saqabi, B., Numerical treatment of fractional heat equations, Appl. numer. math., 58, 1212-1223, (2008) · Zbl 1143.65105 [12] Lin, Y.; Xu, C., Finite difference/spectral approximations for the time-fractional diffusion equation, J. comput. phys., 225, 1533-1552, (2007) · Zbl 1126.65121 [13] Thomée, V., Galerkin finite element methods for parabolic problems, (2006), Springer-Verlag Berlin · Zbl 1105.65102 [14] Ma, J.; Jiang, Y.; Xiang, K., Numerical simulation of blowup in nonlocal reaction – diffusion equations using a moving mesh method, J. comput. appl. math., 230, 8-21, (2009) · Zbl 1166.65067
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.