×

zbMATH — the first resource for mathematics

Superconvergence error estimate of a finite element method on nonuniform time meshes for reaction-subdiffusion equations. (English) Zbl 07259465
Summary: In this paper, we consider superconvergence error estimates of finite element method approximation of Caputo’s time fractional reaction-subdiffusion equations under nonuniform time meshes. For the standard Galerkin method we see that the optimal order error estimate of temporal direction cannot be derived from the weak formulation of the problem. We establish a time-space error splitting argument, which are called the temporal error and the spatial error, respectively. The temporal error is proved skillfully based on an improved discrete Grönwall inequality. We obtain the sharp temporal \(H^1\)-norm error estimates with respect to the convergence order of the approximate solution and \(H^1\)-norm superclose results are given in details. Furthermore, by virtue of the interpolated postprocessing techniques, the global \(H^1\)-norm superconvergence results are presented. Finally, we present some numerical results that give insight into the reliability of the theoretical analysis.
MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35R11 Fractional partial differential equations
26A33 Fractional derivatives and integrals
Software:
FODE
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alikhanov, AA, A new difference scheme for the time fractional diffusion equation, J. Comput. Phys., 280, 424-438 (2015) · Zbl 1349.65261
[2] Chen, H.; Stynes, M., Error analysis of a second-order method on fitted meshes for a time-fractional diffusion problem, J. Sci. Comput., 79, 624-647 (2019) · Zbl 1419.65010
[3] Ciarlet, PG; Lions, JL, Handbook of Numerical Analysis (1991), Amsterdam: North-Holland, Amsterdam
[4] Deng, WH, Finite element method for the space and time fractional Fokker-Planck equation, SIAM J. Numer. Anal., 47, 204-226 (2008) · Zbl 1416.65344
[5] Diethelm, K., The Analysis of Fractional Differential Equations (2004), Berlin: Springer, Berlin
[6] Ji, BQ; Liao, H-L; Zhang, LM, Simple maximum-principle preserving time-stepping methods for time-fractional Allen-Cahn equation, Adv. Comput. Math. (2020) · Zbl 1437.35683
[7] Jiang, Y.; Ma, JT, High-order finite element methods for time-fractional partial differential equations, J. Comput. Appl. Math., 235, 3285-3290 (2011) · Zbl 1216.65130
[8] Jin, B.; Lazarov, R.; Pascal, J.; Zhou, Z., Error analysis of semidiscrete finite element methods for inhomogeneous time-fractional diffusion, IMA J. Numer. Anal., 35, 561-582 (2015) · Zbl 1321.65142
[9] Jin, B.; Lazarov, R.; Zhou, Z., An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. Numer. Anal., 36, 197-221 (2016) · Zbl 1336.65150
[10] Jin, B.; Lazarov, R.; Zhou, Z., Error estimates for a semidiscrete finite element method for fractional order parabolic equations, SIAM J. Numer. Anal., 51, 445-466 (2013) · Zbl 1268.65126
[11] Jin, B.; Lazarov, R.; Zhou, Z., The Galerkin finite element method for a multi-term time-fractional diffusion equation, J. Comput. Phys., 281, 825-843 (2015) · Zbl 1352.65350
[12] Jin, B.; Lazarov, R.; Zhou, Z., Two fully discrete schemes for fractional diffusion and diffusion-wave equations, SIAM J. Sci. Comput., 38, A146-A170 (2016) · Zbl 1381.65082
[13] Li, DF; Liao, H-L; Sun, W.; Wang, J.; Zhang, JW, Analysis of L1-Galerkin FEMs for time-fractional nonlinear parabolic problems, Commun. Comput. Phys., 24, 86-103 (2018)
[14] Li, DF; Wang, J.; Zhang, JW, Unconditionally convergent L1-Galerkin FEMs for nonlinear time-fractional Schrödinger equations, SIAM J. Sci. Comput., 39, A3067-A3088 (2017) · Zbl 1379.65079
[15] Li, DF; Wu, C.; Zhang, ZM, Linearized Galerkin FEMs for nonlinear time fractional Parabolic problems with non-smooth solutions in time direction, J. Sci. Comput., 80, 403-419 (2019) · Zbl 1418.65179
[16] Li, X.; Zhang, L.; Liao, H-L, Sharp \(H^1\)-norm error estimate of a cosine pseudo-spectral scheme for 2D reaction-subdiffusion equations, Numer. Algor., 83, 1223-1248 (2020) · Zbl 1440.65092
[17] Li, Z.; Wang, H.; Yang, DP, A space-time fractional phase-field model with tunable sharpness and decay behavior and its efficient numerical simulation, J. Comput. Phys., 347, 20-38 (2017) · Zbl 1380.65306
[18] Liao, H-L; Li, DF; Zhang, JW, Sharp error estimate of nonuniform L1 formula for linear reaction-subdiffusion equations, SIAM J. Numer. Anal., 56, 1112-1133 (2018) · Zbl 1447.65026
[19] Liao, H-L; McLean, W.; Zhang, JW, A discrete Grönwall inequality with application to numerical schemes for subdiffusion problems, SIAM J. Numer. Anal., 57, 218-237 (2019) · Zbl 1414.65008
[20] Liao, H.-L., McLean, W., Zhang, J.W.: A second-order scheme with nonuniform time steps for a linear reaction-subdiffusion equation (2018). arXiv: 1803.09873v2
[21] Liao, H-L; Yan, Y.; Zhang, JW, Unconditional convergence of a two-level linearized fast algorithm for semilinear subdiffusion equations, J. Sci. Comput., 80, 1-25 (2019) · Zbl 1450.65078
[22] Lin, Q.; Yan, NN, The Construction and Analysis of High Efficient Elements (1996), Hebei: Hebei University Press, Hebei
[23] Lin, YM; Xu, CJ, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225, 1533-1552 (2007) · Zbl 1126.65121
[24] Liu, H.; Cheng, AJ; Wang, H.; Zhao, J., Time-fractional Allen-Cahn and Cahn-Hilliard phase-field models and their numerical investigation, Comput. Math. Appl., 76, 1876-1892 (2018)
[25] Mainardi, F., Fractals and Fractional Calculus Continuum Mechanics (1997), Berlin: Springer, Berlin · Zbl 0917.73004
[26] McLean, W., Regularity of solutions to a time-fractional diffusion equation, ANZIAM J., 52, 123-138 (2010) · Zbl 1228.35266
[27] Oldham, K.; Spanier, J., The Fractional Calculus (1974), New York: Academic Press, New York · Zbl 0428.26004
[28] Podlubny, I., Fractional Differential Equations (1999), New York: Academic Press, New York · Zbl 0918.34010
[29] Ren, JC; Long, XN; Mao, SP; Zhang, JW, Superconvergence of finite element approximations for the fractional diffusion-wave equation, J. Sci. Comput., 72, 917-935 (2017) · Zbl 1397.65162
[30] Ren, J.C., Liao, H.-L., Zhang, J.W., Zhang, Z.M.: Sharp \(H^1\)-norm error estimates of two time-stepping schemes for reaction-subdiffusion problems (2018). arXiv: 1811.08059v1
[31] Ren, JC; Shi, DY; Vong, SW, High accuracy error estimates of a Galerkin FEM for nonlinear time fractional diffusion equation, Numer. Methods Part. Differ. Equ., 36, 284-301 (2020)
[32] Ren, JC; Sun, ZZ; Zhao, X., Compact difference scheme for the fractional sub-diffusion equation with Neumann boundary conditions, J. Comput. Phys., 232, 456-467 (2013) · Zbl 1291.35428
[33] Ren, JC; Sun, ZZ, Numerical algorithm with high spatial accuracy for the fractional diffusion-wave equation with Neumann boundary conditions, J. Sci. Comput., 56, 381-408 (2013) · Zbl 1281.65113
[34] Sakamoto, K.; Yamamoto, M., Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382, 426-447 (2011) · Zbl 1219.35367
[35] Shi, DY; Wang, PL; Zhao, YM, Superconvergence analysis of anisotropic linear triangular finite element for nonlinear Schrödinger equation, Appl. Math. Lett., 38, 129-134 (2014) · Zbl 1314.65127
[36] Stynes, M.; O’Riordan, E.; Gracia, JL, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal., 55, 1057-1079 (2017) · Zbl 1362.65089
[37] Sun, ZZ; Wu, XN, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 56, 193-209 (2006) · Zbl 1094.65083
[38] Thomée, V., Galerkin Finite Element Methods for Parabolic Problems (1997), Berlin: Springer, Berlin · Zbl 0884.65097
[39] Yan, Y.; Khan, M.; Ford, N., An analysis of the modified L1 scheme for time-fractional partial differential equations with nonsmooth data, SIAM J. Numer. Anal., 56, 210-227 (2018) · Zbl 1381.65070
[40] Zhao, YM; Chen, P.; Bu, WP; Liu, XT; Tang, YF, Two mixed finite element methods for time-fractional diffusion equations, J. Sci. Comput., 70, 407-428 (2017) · Zbl 1360.65245
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.