A space-time finite element method for fractional wave problems
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A space-time finite element method for fractional wave problems Binjie Li1 · Hao Luo1
· Xiaoping Xie1
Received: 5 May 2019 / Accepted: 18 November 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we analyze a space-time finite element method for fractional wave problems involving the time fractional derivative of order γ (1 < γ < 2). We first establish the stability of the proposed method and then derive the optimal convergence rate in H 1 (0, T ; L2 ())-norm and suboptimal rate in discrete L∞ (0, T ; H01 ())norm. Furthermore, we discuss the performance of this method when the true solution has singularity at t = 0 and show that optimal convergence rate with respect to H 1 (0, T ; L2 ())-norm can still be achieved by using graded temporal grids. Finally, numerical experiments are performed to verify the theoretical results. Keywords Fractional wave problem · Space-time finite element · Convergence · Singularity · Graded grid
1 Introduction This paper considers the following fractional wave problem: ⎧ γ D (u − u0 − tu1 ) − u = f in × (0, T ), ⎪ ⎪ ⎨ 0+ u = 0 on ∂ × (0, T ), u(·, 0) = u0 in , ⎪ ⎪ ⎩ = u1 in , ut (·, 0)
Hao Luo
[email protected] Binjie Li [email protected] Xiaoping Xie [email protected] 1
School of Mathematics, Sichuan University, Chengdu, 610064, China
(1)
Numerical Algorithms
where 1 < γ < 2, ⊂ Rd (d = 2, 3) is a polygon/polyhedron, and u0 , u1 , and f are given functions. Here ut is the derivative of u with respect to the time variable t, γ and D0+ is a Riemann–Liouville fractional differential operator of order γ . Over the last two decades, the numerical treatment to time fractional partial differential equations has been an active research area. The main difference of these numerical methods is how to discretize the fractional derivatives. So far, there are three types of approaches to discretize the fractional derivatives: finite difference methods, spectral methods, and finite element methods. For the first type of algorithms that use the finite difference methods to discretize the fractional derivatives, we refer the reader to [3, 7, 9–12, 19, 21, 30, 33, 36–38] and the references therein. These algorithms are easy to implement, but are generally of low temporal accuracy. For the second type of algorithms based on the spectral methods, we refer the reader to [16, 20, 39, 41–44]. These algorithms have high-order accuracy if the solution is sufficiently regular. Since singularity is an important feature of time fractional problems, the high-order accuracy of these algorithms is limited. Besides, these algorithms often lead to large scale dense systems to solve. For the third type of algorithms that use the finite element method to discretize fractional derivatives, we refer the reader to [13–15, 17, 18, 22–28]. Similar to the first type of algorithms, the discrete systems arising from these algorithms are solved successively in the time direction. Furthermore, these algorithms possess high-order accuracy, and if the sol
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