[1] |
Ciarlet, P., The Finite Element Method for Elliptic Problems (2002), Philadelphia, PA: SIAM, Philadelphia, PA · doi:10.1137/1.9780898719208 |
[2] |
Eriksson, K.; Johnson, C.; Thomée, V., Time discretization of parabolic problems by the discontinuous Galerkin method, RAIRO Modél. Math. Anal. Numér., 19, 4, 611-643 (1985) · Zbl 0589.65070 · doi:10.1051/m2an/1985190406111 |
[3] |
Ervin, V.; Roop, J., Variational formulation for the stationary fractional advection dispersion equation, Numer. Meth. Part. D. E., 22, 3, 558-576 (2006) · Zbl 1095.65118 · doi:10.1002/num.20112 |
[4] |
Evans, L.: Partial Differential Equations, 2nd Ed. American Mathematical Society (2010) · Zbl 1194.35001 |
[5] |
Ford, N.; Xiao, J.; Yan, Y., A finite element method for time fractional partial differential equations, Fract. Calc. Appl. Anal., 14, 3, 454-474 (2018) · Zbl 1273.65142 · doi:10.2478/s13540-011-0028-2 |
[6] |
Gao, G.; Sun, Z., A compact finite difference scheme for the fractional sub-diffusion equations, J. Comput. Phys., 230, 3, 586-595 (2011) · Zbl 1211.65112 · doi:10.1016/j.jcp.2010.10.007 |
[7] |
Gao, G.; Sun, H.; Sun, Z., Stability and convergence of finite difference schemes for a class of time-fractional sub-diffusion equations based on certain superconvergence, J. Comput. Phys., 280, 510-528 (2015) · Zbl 1349.65295 · doi:10.1016/j.jcp.2014.09.033 |
[8] |
Gorenflo, R., Kilbas, A., Mainardi, F. and Rogosin, S.: Mittag-Leffler Functions, Related Topics and Applications. Springer Monographs in Mathematics. Springer, Berlin (2014); 2nd Ed. (2020) · Zbl 1309.33001 |
[9] |
Karaa, S.; Mustapha, K.; Pani, A., Optimal error analysis of a FEM for fractional diffusion problems by energy arguments, J. Sci. Comput., 74, 1, 519-535 (2018) · Zbl 1398.65228 · doi:10.1007/s10915-017-0450-7 |
[10] |
Ke, R., Ng, M. and Sun, H.: A fast direct method for block triangular Toeplitz-like with tri-diagonal block systems from time-fractional partial differential equations. J. Comput. Phys. 303(C), 203-211 (2015) · Zbl 1349.65404 |
[11] |
Kufner, A., Persson, L. and Samko, N.: Weighted Inequalities of Hardy Type. World Scientific Publishing Company (2017) · Zbl 1380.26001 |
[12] |
Langlands, T.; Henry, B., The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comput. Phys., 205, 2, 719-736 (2005) · Zbl 1072.65123 · doi:10.1016/j.jcp.2004.11.025 |
[13] |
Li, B.; Luo, H.; Xie, X., A space-time finite element method for fractional wave problems, Numer. Algor., 85, 3, 1095-1121 (2020) · Zbl 1451.65149 · doi:10.1007/s11075-019-00857-w |
[14] |
Li, B.; Luo, H.; Xie, X., Analysis of a time-stepping scheme for time fractional diffusion problems with nonsmooth data, SIAM J. Numer. Anal., 57, 2, 779-798 (2019) · Zbl 1419.65066 · doi:10.1137/18M118414X |
[15] |
Li, B.; Wang, T.; Xie, X., Analysis of the L1 scheme for fractional wave equations with nonsmooth data, Comput. Math. Appl., 90, 1-12 (2021) · Zbl 1524.65563 · doi:10.1016/j.camwa.2021.03.006 |
[16] |
Li, B.; Wang, T.; Xie, X., Numerical analysis of a semilinear fractional diffusion equation, Comput. Math. Appl., 80, 2115-2134 (2020) · Zbl 1454.65035 · doi:10.1016/j.camwa.2020.09.008 |
[17] |
Li, B.; Wang, T.; Xie, X., Numerical analysis of two Galerkin discretizations with graded temporal grids for fractional evolution equations, J. Sci. Comput., 85, 3, 85-59 (2020) · Zbl 1456.65116 · doi:10.1007/s10915-020-01365-z |
[18] |
Li, B.; Xie, X., Regularity of solutions to time fractional diffusion equations, Discrete Contin. Dyn. Syst. -B., 24, 3195-3210 (2019) · Zbl 1428.35666 |
[19] |
Li, X.; Xu, C., A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47, 3, 2108-2131 (2009) · Zbl 1193.35243 · doi:10.1137/080718942 |
[20] |
Li, Z.; Yan, Y., Error estimates of high-order numerical methods for solving time fractional partial differential equations, Fract. Calc. Appl. Anal., 21, 3, 746-774 (2018) · Zbl 1405.65098 · doi:10.1515/fca-2018-0039 |
[21] |
Lions, J.; Magenes, E., Non-Homogeneous Boundary Value Problems and Applications (1972), Berlin: Springer, Berlin · Zbl 0223.35039 · doi:10.1007/978-3-642-65217-2 |
[22] |
Luo, H.; Li, B.; Xie, X., Convergence analysis of a Petrov-Galerkin method for fractional wave problems with nonsmooth data, J. Sci. Comput., 80, 2, 957-992 (2019) · Zbl 1477.65161 · doi:10.1007/s10915-019-00962-x |
[23] |
McLean, W., Regularity of solutions to a time-fractional diffusion equation, ANZIAM J., 52, 2, 123-138 (2010) · Zbl 1228.35266 · doi:10.1017/S1446181111000617 |
[24] |
McLean, W.; Mustapha, K., Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation, Numer. Algor., 52, 1, 69-88 (2009) · Zbl 1177.65194 · doi:10.1007/s11075-008-9258-8 |
[25] |
McLean, W., Mustapha, K.: Time-stepping error bounds for fractional diffusion problems with non-smooth initial data. J. Comput. Phys. 293(C), 201-217 (2015) · Zbl 1349.65469 |
[26] |
McLean, W.; Thomée, V., Maximum-norm error analysis of a numerical solution via Laplace transformation and quadrature of a fractional order evolution, IMA J. Numer. Anal., 30, 208-230 (2010) · Zbl 1416.65381 · doi:10.1093/imanum/drp004 |
[27] |
McLean, W.; Thomée, V., Numerical solution via Laplace transforms of a fractional order evolution equation, J. Integral Equ. Appl., 22, 1, 57-94 (2010) · Zbl 1195.65122 · doi:10.1216/JIE-2010-22-1-57 |
[28] |
Mohebbi, A.; Abbaszadeh, M.; Dehghan, M., A high-order and unconditionally stable scheme for the modified anomalous fractional sub-diffusion equation with a nonlinear source term, J. Comput. Phys., 240, 1, 36-48 (2013) · Zbl 1287.65064 · doi:10.1016/j.jcp.2012.11.052 |
[29] |
Mustapha, K., Time-stepping discontinuous Galerkin methods for fractional diffusion problems, Numer. Math., 130, 3, 497-516 (2015) · Zbl 1320.65144 · doi:10.1007/s00211-014-0669-2 |
[30] |
Mustapha, K.; Abdallah, B.; Furati, K., A discontinuous Petrov-Galerkin method for time-fractional diffusion equations, SIAM J. Numer. Anal., 52, 5, 2512-2529 (2014) · Zbl 1323.65109 · doi:10.1137/140952107 |
[31] |
Mustapha, K.; McLean, W., Piecewise-linear, discontinuous Galerkin method for a fractional diffusion equation, Numer. Algor., 56, 2, 159-184 (2011) · Zbl 1211.65127 · doi:10.1007/s11075-010-9379-8 |
[32] |
Mustapha, K.; McLean, W., Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations, SIAM J. Numer. Anal., 51, 1, 491-515 (2013) · Zbl 1267.26005 · doi:10.1137/120880719 |
[33] |
Mustapha, K.; McLean, W., Uniform convergence for a discontinuous Galerkin, time-stepping method applied to a fractional diffusion equation, IMA J. Numer. Anal., 32, 3, 906-925 (2012) · Zbl 1327.65177 · doi:10.1093/imanum/drr027 |
[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, 1, 426-447 (2011) · Zbl 1219.35367 · doi:10.1016/j.jmaa.2011.04.058 |
[35] |
Samko, S.; Kilbas, A.; Marichev, O., Fractional Integrals and Derivatives: Theory and Applications (1993), Switzerland-USA etc: Gordon and Breach Science Publishers, Switzerland-USA etc · Zbl 0818.26003 |
[36] |
Schötzau, D.; Schwab, C., Time discretization of parabolic problems by the \(hp\)-version of the discontinuous Galerkin finite element method, SIAM J. Numer. Anal., 38, 3, 837-875 (2000) · Zbl 0978.65091 · doi:10.1137/S0036142999352394 |
[37] |
Stynes, M., Too much regularity may force too much uniqueness, Fract. Calc. Appl. Anal., 19, 6, 1554-1562 (2016) · Zbl 1353.35306 · doi:10.1515/fca-2016-0080 |
[38] |
Stynes, M.; O’Riordan, E.; Gracia, J., Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal., 55, 2, 1057-1079 (2017) · Zbl 1362.65089 · doi:10.1137/16M1082329 |
[39] |
Tartar, L., An Introduction to Sobolev Spaces and Interpolation Spaces (2007), Berlin: Springer, Berlin · Zbl 1126.46001 |
[40] |
Thomée, V., Galerkin Finite Element Methods for Parabolic Problems (2006), Berlin: Springer, Berlin · Zbl 1105.65102 |
[41] |
Wang, Y.; Yan, Y.; Yang, Y., Two high-order time discretization schemes for subdiffusion problems with nonsmooth data, Fract. Calc. Appl. Anal., 23, 5, 1349-1380 (2020) · Zbl 1474.65293 · doi:10.1515/fca-2020-0067 |
[42] |
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, 1, 210-227 (2018) · Zbl 1381.65070 · doi:10.1137/16M1094257 |
[43] |
Yang, Y.; Yan, Y.; Ford, N., Some time stepping methods for fractional diffusion problems with nonsmooth data, Comput. Methods Appl. Math., 18, 1, 129-146 (2018) · Zbl 1383.65097 · doi:10.1515/cmam-2017-0037 |
[44] |
Yuste, S., Weighted average finite difference methods for fractional diffusion equations, J. Comput. Phys., 216, 264-274 (2006) · Zbl 1094.65085 · doi:10.1016/j.jcp.2005.12.006 |
[45] |
Yuste, S.; Acedo, L., An explicit finite difference method and a new von-Neumann-type stability analysis for fractional diffusion equations, SIAM J. Numer. Anal., 42, 5, 1862-1874 (2005) · Zbl 1119.65379 · doi:10.1137/030602666 |
[46] |
Zhuang, P.; Liu, F.; Anh, V.; Turner, I., New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation, SIAM J. Numer. Anal., 46, 2, 1079-1095 (2008) · Zbl 1173.26006 · doi:10.1137/060673114 |