Numerical Analysis of Two Galerkin Discretizations with Graded Temporal Grids for Fractional Evolution Equations
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Numerical Analysis of Two Galerkin Discretizations with Graded Temporal Grids for Fractional Evolution Equations Binjie Li1 · Tao Wang2
· Xiaoping Xie1
Received: 9 March 2020 / Revised: 23 September 2020 / Accepted: 8 November 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Two numerical methods with graded temporal grids are analyzed for fractional evolution equations. One is a low-order discontinuous Galerkin (DG) discretization in the case of fractional order 0 < α < 1, and the other one is a low-order Petrov Galerkin (PG) discretization in the case of fractional order 1 < α < 2. By a new duality technique, pointwise-in-time error estimates of first-order and (3 − α)-order temporal accuracies are respectively derived for DG and PG, under reasonable regularity assumptions on the initial value. Numerical experiments are performed to verify the theoretical results. Keywords Fractional evolution equation · Graded temporal grid · Convergence
1 Introduction Let X be a separable Hilbert space with inner product (·, ·) X . Assume that the linear operator A : D(A) ⊂ X → X is densely defined and admits a bounded inverse A−1 : X → X , which is compact, symmetric and positive. Consider the following time fractional evolution equation:
α D0+ (u − u 0 ) (t) + Au(t) = 0, 0 < t ≤ T ,
(1)
α is a Riemann-Liouville fractional where α ∈ (0, 2) \ {1}, 0 < T < ∞, u 0 ∈ X and D0+ derivative operator of order α. Here, we assume that u(0) = u 0 for α ∈ (0, 2) \ {1} and u (0) = 0 for α ∈ (1, 2).
B
Tao Wang [email protected] Binjie Li [email protected]; [email protected] Xiaoping Xie [email protected]
1
School of Mathematics, Sichuan University, Chengdu, China
2
South China Reasearh Center for Applied Mathmatics and Interdisciplinary Studies, South China Normal University, Guangzhou, China 0123456789().: V,-vol
123
59
Page 2 of 27
Journal of Scientific Computing
(2020) 85:59
There are quite a few research works on the numerical treatment of time fractional evolution equations. Let us briefly introduce four types of numerical methods for the discretization of time fractional evolution equations. The first-type method uses the convolution quadrature to approximate the fractional integral (derivative) (cf. [2,6,16,17,36]). The second-type method uses the L1 scheme to approximate the fractional derivative (cf. [3,5,12,15,31,32]). Such methods are popular and easy to implement. The third-type method is the spectral method (cf. [9,14,20,33,34]), which uses nonlocal basis functions to approximate the solution. The accuracy of the spectral method is high, provided that the solution or data is smooth enough. The fourth-type method is the finite element method (cf. [10,13,19,21,24,25]), which uses local basis functions to approximate the solution. It should be mentioned that the finite element method is identical to the L1 scheme in some cases (cf. [7,12]). Most of the convergence analyses for the numerical methods mentioned above are based on the assumption that the exact soluti
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