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Isogeometric analysis for time-fractional partial differential equations. (English) Zbl 1450.65123
Summary: We consider isogeometric analysis to solve the time-fractional partial differential equations: fractional diffusion and diffusion-wave equations. Traditional spatial discretization for time-fractional models include finite differences, finte elements, spectral methods, etc. A novel method-isogeometric analysis is used for spatial discretization in this paper. The traditional \(L1\) scheme and \(L2\) scheme are used for time discretization of our models. Isogeometric analysis has potential advantages in exact geometry representations, efficient mesh generation, \(h\)- and \(k\)- refinements, and smooth basis functions. We show stability and a priori error estimates for spatial discretization and the space-time fully discrete scheme. A variety of numerical examples in 2d and 3d are provided to verify theory and show accuracy, efficiency, and convergence of isogeometric analysis based on B-splines and non-uniform rational B-splines.
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65D07 Numerical computation using splines
26A33 Fractional derivatives and integrals
74S05 Finite element methods applied to problems in solid mechanics
35R11 Fractional partial differential equations
Full Text: DOI
[1] Bazilevs, Y.; Beirão de Veiga, L.; Cottrell, JA; Hughes, TJR; Sangalli, G., Isogeometric analysis: Approximation, stability and error estimates for h-refined meshes, Math. Models Methods Appl. Sci., 16, 1031-1090 (2006) · Zbl 1103.65113
[2] Beirão de Veiga, L.; Buffa, A.; Sangalli, G.; Vazquez, R., Mathematical analysis of variational isogeometric methods, Acta Numer., 23, 157-287 (2014) · Zbl 1398.65287
[3] Chen, F.; Xu, Q.; Hesthaven, JS, A multi-domain spectral method for time-fractional differential equations, J. Comput. Phys., 293, 157-172 (2015) · Zbl 1349.65506
[4] Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric analysis: toward integration of CAD and FEA. Wiley (2009) · Zbl 1378.65009
[5] Dai, P.; Wu, Q.; Zhu, S., Quasi-Toeplitz splitting iteration methods for unsteady space-fractional diffusion equations, Numer. Methods Partial Diff. Equ., 35, 699-715 (2019) · Zbl 1418.65097
[6] Deng, W., Numerical algorithm for the time fractional Fokker-Planck equation, J. Comput. Phys., 227, 1510-1522 (2007) · Zbl 1388.35095
[7] de Falco, C.; Reali, A.; Vàzquez, R., GeoPDEs: a research tool for isogeometric analysis of PDEs, Adv. Eng. Softw., 42, 1020-1034 (2011) · Zbl 1246.35010
[8] Feng, LB; Liu, P.; Zhuang, F.; Turner, I., Stability and convergence of a new finite volume method for a two-sided space-fractional diffusion equation, Appl. Math. Comput., 257, 52-65 (2015) · Zbl 1339.65144
[9] Feng, LB; Zhuang, P.; Liu, F.; Turner, I.; Gu, YT, Finite element method for space-time fractional diffusion equation, Numer. Algor., 72, 749-767 (2016) · Zbl 1343.65122
[10] 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
[11] Hughes, TJR; Cottrell, J.; Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement, Comput. Methods Appl. Mech. Engrg., 194, 4135-4195 (2005) · Zbl 1151.74419
[12] 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
[13] 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
[14] 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
[15] Li, C.; Zhao, Z.; Chen, Y., Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Comput. Math. Appl., 62, 855-875 (2011) · Zbl 1228.65190
[16] Li, R.; Wu, Q.; Zhu, S., Proper orthogonal decomposition with SUPG-stabilized isogeometric analysis for reduced order modelling of unsteady convection-dominated convection-diffusion-reaction problems, J. Comput. Phys., 387, 280-302 (2019)
[17] 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, 488-504 (2005)
[18] Lin, Y.; Xu, C., Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225, 1533-1552 (2007) · Zbl 1126.65121
[19] Mainardi, F., Fractional calculus and waves in linear viscoelasticity. An introduction to mathematical models (2010), London: Imperial College Press, London · Zbl 1210.26004
[20] Mustapha, K.; Abdallah, B.; Furati, KM, A discontinuous Petrov-Galerkin method for time-fractional diffusion equations, SIAM J. Numer. Anal., 52, 2512-2529 (2014) · Zbl 1323.65109
[21] Piegl, L.; Tiller, W., The NURBS Book (1997), New York: Springer, New York
[22] Podlubny, I., Fractional differential equations (1999), San Diego: Academic Press, San Diego · Zbl 0918.34010
[23] Quarteroni, A.; Valli, A., Numerical approximation of partial differential problems (1997), Berlin: Springer-Verlag, Berlin
[24] 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
[25] Sun, Z.; Wu, X., A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 56, 193-209 (2006) · Zbl 1094.65083
[26] Thomée, V., Galerkin finite element methods for parabolic problems (1997), Berlin: Springer, Berlin · Zbl 0884.65097
[27] Wang, H.; Cheng, A.; Wang, K., Fast finite volume methods for space-fractional diffusion equations, Discrete Contin. Dyn. Syst. Ser. B, 20, 1427-1441 (2015) · Zbl 1382.65272
[28] Wang, H.; Yang, D.; Zhu, S., Inhomogeneous Dirichlet boundary-value problems of space-fractional diffusion equations and their finite element approximations, SIAM J. Numer. Anal., 52, 1292-1310 (2014) · Zbl 1320.65182
[29] Wang, H.; Yang, D.; Zhu, S., A Petrov-Galerkin finite element method for variable-coefficient fractional diffusion equations, Comput. Methods Appl. Mech. Engrg., 290, 45-56 (2015) · Zbl 1425.65183
[30] Wang, H.; Yang, D.; Zhu, S., Accuracy of finite element methods for boundary-value problems of steady-state fractional diffusion equations, J. Sci. Comput., 70, 429-449 (2017) · Zbl 1359.65271
[31] Zeng, F.; Li, C.; Liu, F.; Turner, I., The use of finite difference/element approaches for solving the time-fractional subdiffusion equation, SIAM J. Sci. Comput., 35, A2976-A3000 (2013) · Zbl 1292.65096
[32] Zeng, F.; Zhang, Z.; Karniadakis, GE, Fast difference schemes for solving high-dimensional time-fractional subdiffusion equations, J. Comput. Phys., 307, 15-33 (2016) · Zbl 1352.65278
[33] Zayernouri, M.; Karniadakis, GE, Discontinuous spectral element methods for time-and space-fractional advection equations, SIAM J. Sci. Comput., 36, B684-B707 (2014) · Zbl 1304.35757
[34] 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
[35] Zhu, S.; Dedè, L.; Quarteroni, A., Isogeometric analysis and proper orthogonal decomposition for parabolic problems, Numer. Math., 135, 333-370 (2017) · Zbl 1380.65295
[36] Zhu, S.; Dedè, L.; Quarteroni, A., Isogeometric analysis and proper orthogonal decomposition for the acoustic wave equation, ESAIM Math. Model. Numer. Anal., 51, 1197-1221 (2017) · Zbl 1381.65086
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