Convergence and superconvergence analyses of HDG methods for time fractional diffusion problems
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Convergence and superconvergence analyses of HDG methods for time fractional diffusion problems Kassem Mustapha1 · Maher Nour1 · Bernardo Cockburn2
Received: 17 November 2014 / Accepted: 1 October 2015 © Springer Science+Business Media New York 2015
Abstract We study the hybridizable discontinuous Galerkin (HDG) method for the spatial discretization of time fractional diffusion models with Caputo derivative of order 0 < α < 1. For each time t ∈ [0, T], when the HDG approximations are taken to be piecewise polynomials of degree k ≥ 0 on the spatial domain , the approximations to the exact solution u in the L∞ (0, T; L2 ())-norm and to ∇u in the L∞ (0, T; L2 ())-norm are proven to converge with the rate hk+1 provided that u is sufficiently regular, where h is the maximum diameter of the elements of the mesh. Moreover, for k ≥ 1, we obtain a superconvergence result which allows us to compute, in an elementwise manner, a new approximation for u converging with a rate hk+2 (ignoring the logarithmic factor), for quasi-uniform spatial meshes. Numerical experiments validating the theoretical results are displayed. Keywords Anomalous diffusion · Time fractional · Discontinuous Galerkin methods · Hybridization · Convergence analysis Communicated by: Jan Hesthaven Support of the King Fahd University of Petroleum and Minerals (KFUPM) through the project FT131011 is gratefully acknowledged. Kassem Mustapha
[email protected] Maher Nour [email protected] Bernardo Cockburn [email protected] 1
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
2
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
K. Mustapha et al.
Mathematics Subject Classification (2010) 26A33 · 65M12 · 65M15 · 65N30
1 Introduction In this paper, we study the method resulting from using exact integration in time and a hybridizable discontinuous Galerkin (HDG) method for the spatial discretization of the following time fractional diffusion model problem: D1−α u(x, t) − u(x, t) = f (x, t) for (x, t) ∈ × (0, T ], u(x, t) = g(x) for (x, t) ∈ ∂ × (0, T ],
c
(1a) (1b)
with u(x, 0) = u0 (x) for x ∈ , where is a convex polyhedral domain of Rd (d = 1, 2, 3) with boundary ∂, f, g and u0 are given functions assumed to be sufficiently regular such that the solution u of Eq. 1 is in the space W 1,1 (0, T ; H 2 ()), (further regularity assumptions will be imposed later), and T > 0 is a fixed but arbitrary value. Here, c D1−α denotes time fractional Caputo derivative of order α defined by t c 1−α α D v(t) := I v (t) := ωα (t − s)v (s) ds with 0 < α < 1, (2) 0
where v denotes the time derivative of the function v and I α is the Riemann– t α−1 Liouville (time) fractional integral operator; with ωα (t) := (α) and being the gamma function. In this work, we investigate a high-order accurate numerical method for the space discretization for problem (1). Using exact integration in time, we propose to deal with the accuracy issue by developing a high-o
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