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A new gradient scheme of a time fractional Fokker-Planck equation with time independent forcing and its convergence analysis. (English) Zbl 07239613
Klöfkorn, Robert (ed.) et al., Finite volumes for complex applications IX – methods, theoretical aspects, examples. FVCA 9, Bergen, Norway, June 15–19, 2020. In 2 volumes. Volume I and II. Cham: Springer (ISBN 978-3-030-43650-6/hbk; 978-3-030-43651-3/ebook). Springer Proceedings in Mathematics & Statistics 323, 285-293 (2020).
Summary: We apply the GDM (Gradient Discretization Method) developed recently in [5, 6] to approximate the time fractional Fokker-Planck equation with time independent forcing in any space dimension. Using [5] which dealt with GDM for linear advection problems, we develop a new fully discrete implicit GS (Gradient Scheme) for the stated model. We prove new discrete a priori estimates which yield estimates on the discrete solution in \(L^\infty (L^2)\) and \(L^2(H^1)\) discrete norms. Thanks to these discrete a priori estimates, we prove new error estimates in the discrete norms of \(L^\infty (L^2)\) and \(L^2(H^1)\). The main ingredients in the proof of these error estimates are the use of the stated discrete a priori estimates and a comparison with some well chosen auxiliary schemes. These auxiliary schemes are approximations of convective-diffusive elliptic problems in each time level. We state without proof the convergence analysis of these auxiliary schemes. Such proof uses some adaptations of the [6, Proof of Theorem 2.28] dealt with GDM for the case of elliptic diffusion problems. These results hold for all the schemes within the framework of GDM. This work can be viewed as an extension to our recent one [2].
For the entire collection see [Zbl 1445.65003].
MSC:
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
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[1] Alnashri, Y., Droniou, J.: A gradient discretization method to analyze numerical schemes for nonlinear variational inequalities, application to the seepage problem. SIAM J. Numer. Anal. 56(4), 2375-2405 (2018) · Zbl 1397.35108
[2] Bradji, A.: A new analysis for the convergence of the gradient discretization method for multidimensional time fractional diffusion and diffusion-wave equations. Comput. Math. Appl. 79(2), 500-520 (2020)
[3] Bradji, A.: Notes on the convergence order of gradient schemes for time fractional differential equations. C. R. Math. Acad. Sci. Paris 356(4), 439-448 (2018) · Zbl 1447.65071
[4] Deng, W.: Finite element method for the space and time fractional Fokker-Planck equation. SIAM J. Numer. Anal. 47(1), 204-226 (2008/2009) · Zbl 1416.65344
[5] Droniou, J., Eymard, R., Gallouët, T., Herbin, R.: The Gradient discretisation method for linear advection problems. Comput. Methods Appl. Math. https://doi.org/10.1515/cmam-2019-0060. Accessed 17 Oct 2019 · Zbl 1435.65005
[6] Droniou, J., Eymard, R., Gallouët, T., Guichard, C., Herbin, R.: The Gradient Discretisation Method. Mathématiques et Applications, vol. 82. Springer Nature Switzerland AG, Switzerland (2018) · Zbl 1435.65005
[7] Le, K.N., McLean, W., Mustapha, K.: A semidiscrete finite element approximation of a time-fractional Fokker-Planck equation with nonsmooth initial data. SIAM J. Sci. Comput. 40(6), A3831-A3852 (2018) · Zbl 1404.65123
[8] Pinto, L.
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