Abstract
We propose a piecewise-linear, time-stepping discontinuous Galerkin method to solve numerically a time fractional diffusion equation involving Caputo derivative of order μ ∈ (0, 1) with variable coefficients. For the spatial discretization, we apply the standard continuous Galerkin method of total degree ≤ 1 on each spatial mesh elements. Well-posedness of the fully discrete scheme and error analysis will be shown. For a time interval (0, T) and a spatial domain Ω, our analysis suggest that the error in \(L^{2}\left ((0,T),L^{2}({\Omega })\right )\)-norm is \(O(k^{2-\frac {\mu }{2}}+h^{2})\) (that is, short by order \(\frac {\mu }{2}\) from being optimal in time) where k denotes the maximum time step, and h is the maximum diameter of the elements of the (quasi-uniform) spatial mesh. However, our numerical experiments indicate optimal O(k 2 + h 2) error bound in the stronger \(L^{\infty }\left ((0,T),L^{2}({\Omega })\right )\)-norm. Variable time steps are used to compensate the singularity of the continuous solution near t = 0.
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References
Alikhanov, A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424–438 (2015)
Brunner, H., Ling, L., Yamamoto, M.: Numerical simulations of 2D fractional subdiffusion problems. J. Comput. Phys. 229, 6613–6622 (2010)
Chen, J., Liu, F., Liu, Q., Chen, X., Anh, V., Turner, I., Burrage, K.: Numerical simulation for the three-dimension fractional sub-diffusion equation. Appl. Math. Model. 38, 3695–3705 (2014)
Cui, M.: Compact exponential scheme for the time fractional convection-diffusion reaction equation with variable coefficients. J. Comput. Phys. 280, 143–163 (2013)
Jin, B., Larzarov, R., Zhou, Z.: Error estimates for a semidiscrete FE method for fractional order parabolic equations. SIAM J. Numer. Anal. 51, 445–466 (2013)
Li, W., Xu, D.: Finite central difference/FE approximations for parabolic integro-differential equations. Computing 90, 89–111 (2010)
Li, X., Xu, C.: A Space-Time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47, 2108–2131 (2009)
Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)
Luskin, M., Rannacher, R.: On the smoothing property of the Galerkin method for parabolic equations. SIAM J. Numer. Anal. 19, 93–113 (1982)
Mclean, W.: Regularity of solutions to a time-fractional diffusion equation. ANZIAM J. 52, 123–138 (2010)
Mclean, W.: Fast summation by interval clustering for an evolution equation with memory. SIAM J. Sci. Comput. 34, A3039–A3056 (2012)
Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics Reports 339, 1–77 (2000)
Murillo, J., Yuste, S. B.: A finite difference method with non-uniform timesteps for fractional diffusion and diffusion-wave equations. Eur. Phys. J. Special Topics 222, 1987–1998 (2013)
Mustapha, K., McLean, W.: Piecewise-linear, discontinuous Galerkin method for a fractional diffusion equation. Numer. Algor. 56, 159–184 (2011)
Mustapha, K., McLean, W.: Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations. SIAM J. Numer. Anal. 51, 491–515 (2013)
Mustapha, K.: Time-stepping discontinuous Galerkin methods for fractional diffusion problems. Numer. Math. 130, 497–516 (2015)
Mustapha, K., Abdallah, B., Furati, K. M.: A discontinuous Pertov-Galerkin method for time-fractional diffusion equations. SIAM J. Number. Anal. 52, 2512–2529 (2014)
Mustapha, K., Nour, M., Cockburn, B.: Convergence and superconvergence analyses of HDG methods for time fractional diffusion problems. Adv. Comput. Math. (2015). doi:10.1007/s10444-015-9428-x
Mustapha, K., Schötzau, D.: Well-posedness of hp-version discontinuous Galerkin method for fractional wave equations. IMA J. Numer. Anal. 34, 1426–1446 (2014)
Ren, J., Sun, Z., Zhao, X.: Compact difference scheme for the fractional sub-diffusion equation with Neumann boundary conditions. J. Comput. Phys. 232, 456–467 (2013)
Saadatmandi, A., Dehghan, M., Azizi, M. R.: The Sinc-Legendre collocation method for a class of fractional convection diffusion equations with variable coefficients. Commun. Nonlinear Sci. Numer. Simul. 17, 4125–4136 (2012). doi:10.1007/s10444-015-9428-x
Wang, K. G.: Long time correlation effects and biased anomalous diffusion. Phys. Rev. A 45, 833–837 (1992)
Xu, Q., Zheng, Z.: Discontinuous Galerkin method for time fractional diffusion equation. J. Informat. Comput. Sci. 10, 3253–3264 (2013)
Zhang, Y., Sun, Z., Liao, H. l.: Finite difference methods for the time fractional diffusion equation on non-uniform meshes. J. Comput. Phys. 265, 195–210 (2014)
Zhao, X., Sun, Z.-Z.: A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions. J. Comput. Phys. 230, 6061–6074 (2011)
Zhao, X., Xu, Q.: Efficient numerical schemes for fractional sub-diffusion equation with the spatially variable coefficient. Appl. Math. Model. 38, 3848–3859 (2014)
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The support of the Science Technology Unit at KFUPM through King Abdulaziz City for Science and Technology (KACST) under National Science, Technology and Innovation Plan (NSTIP) project No. 13-MAT1847-04 is gratefully acknowledged.
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Mustapha, K., Abdallah, B., Furati, K.M. et al. A discontinuous Galerkin method for time fractional diffusion equations with variable coefficients. Numer Algor 73, 517–534 (2016). https://doi.org/10.1007/s11075-016-0106-y
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DOI: https://doi.org/10.1007/s11075-016-0106-y