### SUPERCONVERGENCE ANALYSIS FOR TIME-FRACTIONAL DIFFUSION EQUATIONS WITH NONCONFORMING MIXED FINITE ELEMENT METHOD

Houchao Zhang1, Dongyang Shi2

1. 1. School of Mathematics and Statistics, Pingdingshan University, Pingdingshan 467000, China;
2. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China
• Received:2017-10-28 Revised:2018-01-23 Online:2019-07-15 Published:2019-07-15
• Supported by:

The authors are very thankful to the anonymous referees for their helpful suggestions. This work was supported by the National Natural Science Foundation of China (No. 11671369; 11271340).

Houchao Zhang, Dongyang Shi. SUPERCONVERGENCE ANALYSIS FOR TIME-FRACTIONAL DIFFUSION EQUATIONS WITH NONCONFORMING MIXED FINITE ELEMENT METHOD[J]. Journal of Computational Mathematics, 2019, 37(4): 488-505.

In this paper, a fully discrete scheme based on the L1 approximation in temporal direction for the fractional derivative of order in (0, 1) and nonconforming mixed finite element method (MFEM) in spatial direction is established. First, we prove a novel result of the consistency error estimate with order O(h2) of EQ1rot element (see Lemma 2.3). Then, by using the proved character of EQ1rot element, we present the superconvergent estimates for the original variable u in the broken H1-norm and the flux p=∇u in the (L2)2-norm under a weaker regularity of the exact solution. Finally, numerical results are provided to confirm the theoretical analysis.

CLC Number:

 [1] R. Gorenflo, F. Mainardi, D. Moretti, P. Paradisi, Time fractional diffusion:a discrete random walk approach, Nonlinear Dynam., 29(2002), 129-143.[2] R. Gorenflo, Y. Luchko, F. Mainardi, Wright function as scale-invariant solutions of the diffusionwave equation, J. Comp. Appl. Math., 118(2000), 175-191.[3] W. R. Schneider, W. Wyss, Fractional diffusion and wave equations, J. Math. Phys., 30(1989), 134-144.[4] W. Wyss, The fractional diffusion equation, J. Math. Phys., 27(1986), 2782-2785.[5] T. S. Basu, H. Wang, A fast second-order finite difference method for space-fractional diffusion equations, Int. J. Numer. Anal. Modeling, 9(2012), 658-666.[6] M. Cui, Compact finite difference method for the fractional diffusion equation, J. Comput. Phys., 228(2009), 7792-7804.[7] R. Du, W. Cao, Z. Sun, A compact difference scheme for the fractional diffusion-wave equation, Appl. Math. Model., 34(2010), 2998-3007.[8] G. Gao, Z. Sun, A compact finite difference scheme for the fractional sub-diffusion equations, J. Comput. Phys., 230(2011), 586-595.[9] T. A. M. Langlands, B. I. Henry, The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comput. Phys., 205(2005), 719-736.[10] J. Ren, Z. Sun, Numerical algorithm with high spatial accuracy for the fractional diffusion-wave equation with neumann boundary conditions, J. Sci. Comput., 56(2013), 381-408.[11] Z. Sun, X. Wu, A fully discrete scheme for a diffusion wave system, Appl. Numer. Math., 56(2006), 193-209.[12] X. Zhao, Z. Sun, Compact Crank-Nicolson schemes for a class of fractional cattaneo equation in inhomogeneous medium, J. Sci. Comput., 62(2015), 747-771.[13] P. Wang, C. Huang, An energy conservative difference scheme for the nonlinear fractional Schröodinger equations, J. Comput. Phys., 293(2015), 238-251.[14] Y. Jiang, A new analysis of stability and convergence for finite difference schemes solving the time fractional Fokker-Planck equation, Appl. Math. Model., 39(2015), 1163-1171.[15] F. Zeng, C. Li, F. Liu, I. Turner, The use of finite difference/element approaches for solving the time-fractional subdiffusion equation, SIAM J. Sci. Comput., 35(2013), A2976-A3000.[16] H. Wang, T. S. Basu, A fast finite difference method for two-dimensional space-fractional diffusion equations, SIAM J. Sci. Comput., 34(2012), A2444-A2458.[17] V. J. Ervin, N. Heuer, J. P. Roop, Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation, SIAM J. Numer. Anal., 45(2007), 572-591.[18] Y. Jiang, J. Ma, High-order finite element methods for time-fractional partial differential equations, J. Comput. Appl. Math., 235(2011), 3285-3290.[19] 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.[20] B. Jin, R. Lazarov, Z. Zhou, Error estimates for a semidiscrete finite element method for fractional order parabolic equations, SIAM J. Numer. Anal. 51(2013), 445-466.[21] W. Bu, Y. Tang, Y. Wu, J. Yang, Finite difference/finite element method for two-dimensional space and time fractional Bloch-Torrey equations, J. Comput. Phys., 293(2015), 264-279.[22] W. Deng, Finite element method for the space and time fractional Fokker-Planck equation, SIAM J. Numer. Anal., 47(2008), 204-226.[23] H. Wang, D. Yang, S. F. Zhu, Accuracy of finite element methods for boundary-value problems of steady-state fractional diffusion equations, J. Sci. Comput., 70(2017), 429-499.[24] D. Li, H. Liao, W. Sun et al, Analysis of L1-Galerkin FEMs for time-fractional nonlinear parabolic problems, Numer. Anal., arXiv:1612.00562.[25] B. Jin, R. Lazarov, Z. Zhou, Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data, SIAM J. Sci. Comput., 38(2014), A146-A170.[26] X. J. Li, C. J. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47(2009), 2108-2131.[27] Y. Lin, C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225(2007), 1533-1552.[28] C. Lv, C. Xu, Improved error estimates of a finite difference/speciral method for time-fractional diffusion equation, Int. J. Numer. Anal. Model., 12(2015), 384-400.[29] C. Lv, C. Xu, Error analysis of a high order method for time-fractional diffusion equations, SIAM J. Sci. Comput., 38(2016), A2699-A2724.[30] Q. Xu, Z. Zheng, Discontinuous Galerkin method for time fractional diffusion equation, J. Inf. Comput. Sci. 10(2013), 3253-3264.[31] K. Mustapha, Time-stepping discontinuous Galerkin methods for fractional diffusion problems, Numer. Math., 130(2015), 497-516.[32] L. Wei, Y. He, Analysis of a fully discrete local discontinuous Galerkin method for time-fractional fourth-order problems, Appl. Math. Model., 38(2014), 1511-1522.[33] Y. Liu, Y. Du, H. Li, J. Li, S. He, A two-grid mixed finite element method for a nonlinear fourthorder reaction-diffusion problem with time-fractional derivative, Comput. Math. Appl., 70(2015), 2474-2492.[34] Y. Liu, Z. Fang, H. Li, S. He, A mixed finite element method for a time-fractional fourth-order partial differential equation, Appl. Math. Comput., 243(2014), 703-717.[35] X. Zhao, X. Z. Hu, W. Cai, G. E. Karniadakis, Adaptive finite element method for fractional differential equations using Hierarchical Matricesn, Comput. Methods Appl. Mech. Engrg., 325(2017), 56-76.[36] S. Jiang, J. Zhang, Q. Zhang, Z. Zhang, Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations, Commun. Comput. Phys., 21(2017), 650-678.[37] R. Ke, M. Ng, H. Sun, A fast direct method for block triangular Toeplitz-like with tridiagonal block systems for time-fractional partial differential equations, J. Comput. Phys., 303(2015), 203-211.[38] X. Lu, H. Pang, H. Sun, Fast approximate inversion of a block triangular Toeplitz matrix with applications to fractional sub-diffusion equations, Numer. Linear Algebra Appl., 22(2015), 866-882.[39] K. Mustapha, W. Mclean, Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations, SIAM J. Numer. Anal., 51(2013), 491-515.[40] X. Zhao, Z. Zhang, Superconvergence point of fractional spectral interpolation, SIAM J. Sci. Comput., 38(2016), A598-A613.[41] L. Fatone, D. Funaro, Optimal collocation nodes for fractional derivative operators, SIAM J. Sci. Comput., 37(2015), A1504-A1524.[42] Y. Zhao, Y. Zhang, D. Shi, F. Liu, I. Turnerb, Superconvergence analysis of nonconforming finite element method for two-dimensional time fractional diffusion equations, Appl. Math. Lett., 59(2016), 38-47.[43] Z. Hao, G. Lin, Z. Sun, A high-order compact finite difference scheme for the fractional subdiffusion equation, J. Sci. Comput., 64(2015), 959-985.[44] Y. Jiang, J. Ma, Moving finite element methods for time fractional partial differential equations, Sci. China Math., 56(2013), 1287-1300.[45] Y. Liu, Y. Du, H. Li, J. Wang, An H 1-Galerkin mixed finite element method for time fractional reaction-diffusion equation, J. Appl. Math. Comput. 47(2015), 103-117.[46] Y. Zhao, P. Chen, W. Bu, X. Liu, Y. Tang, Two mixed finite element methods for time-fractional diffusion equations, J. Sci. Comput., 70(2017), 407-428.[47] Q. Lin, Tobiska L, A. Zhou, Superconvergence and extrapolation of non-conforming low order finite elements applied to the Poisson equation, IAM J. Numer. Anal., 25(2005), 160-181.[48] D. Shi, S. Mao, S. Chen. An anisotropic nonconforming finite element with some superconvergence results, J. Comput. Math., 23(2005), 261-274.[49] D. Shi, Y. Zhang, High accuracy analysis of a new nonconforming low order finite element scheme for Sobolev equation, Appl. Math. Comput., 218(2011), 3176-3186.[50] Q. Lin, J. Lin, Finite Element Methods:Accuracy and Improvement, Science Press, 2006.[51] Q. Lin, N. Yan, High Efficient Finite Elements (in chinese), Hebei University Press, 1996.[52] Rannacher R, Turek S, Simple nonconforming quadrilateral Stokes element, Numer. Methods Partial Differential Equations, 8(1992), 97-111.[53] J. Hu, Z. Shi, Constrained quadrilateral nonconforming rotated Q1 element, J. Comput. Math., 23(2005), 561-586.[54] M. Stynes, E. O'Riordan, J. L. Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal., 55(2017), 1057-1079.[55] H. Liao, D. Li, J. Zhang, Sharp error estimate of the nonuniform L1 formula for linear reactionsubdiffusion equations, SIAM J. Numer. Anal., (2018), Accepted.
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