Error Estimate of a Fully Discrete Local Discontinuous Galerkin Method for Variable-Order Time-Fractional Diffusion Equa
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Error Estimate of a Fully Discrete Local Discontinuous Galerkin Method for Variable‑Order Time‑Fractional Diffusion Equations Leilei Wei1 · Shuying Zhai2 · Xindong Zhang3 Received: 6 March 2020 / Revised: 13 May 2020 / Accepted: 28 May 2020 © Shanghai University 2020
Abstract The aim of this paper is to develop a fully discrete local discontinuous Galerkin method to solve a class of variable-order fractional diffusion problems. The scheme is discretized by a weighted-shifted Grünwald formula in the temporal discretization and a local discontinuous Galerkin method in the spatial direction. The stability and the L2-convergence of the scheme are proved for all variable-order 𝛼(t) ∈ (0, 1) . The proposed method is of accuracyorder O(𝜏 3 + hk+1 ) , where 𝜏 , h, and k are the temporal step size, the spatial step size, and the degree of piecewise Pk polynomials, respectively. Some numerical tests are provided to illustrate the accuracy and the capability of the scheme. Keywords Variable-order derivative · Discontinuous Galerkin method · Stability · Error estimates Mathematics Subject Classification 65M06
1 Introduction Fractional calculus which is an ancient field and equally important like calculus of integer order is a branch of mathematical analysis that studies the properties of defining real number powers or complex number powers of the differentiation operator [12, 38]. In comparison with the integer-order derivatives, fractional derivative can describe different complex dynamical systems more accurately, since it simultaneously possesses memory, which makes it a powerful tool in modeling physical phenomena related to nonlocality and memory effect [2, 33, 51, 56]. In the past few decades, the fractional calculus has attracted more and more attention in modelling various applications and various high-order numerical
* Leilei Wei [email protected]; [email protected] 1
College of Science, Henan University of Technology, Zhengzhou 450001, China
2
School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, Fujian Province, China
3
College of Mathematics Sciences, Xinjiang Normal University, Urumqi 830054, China
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Communications on Applied Mathematics and Computation
techniques have been developed for the fractional differential systems [3, 4, 7, 9, 19–22, 25, 43]. The variable-order fractional derivatives whose order of fractional derivatives depends on the space and/or time variable are an extension of the classical fractional calculus [1]. Recently, several scholars have paid their attention to the variable-order fractional calculus. Sun et al. [35] discussed the importance of variable-order fractional derivatives. Coimbra [8] studied the variable-order differential operators. In Ref. [15], Evans and Jacob obtained the Feller semigroups by the variable-order subordination. Lorenzo and Hartley [31] investigated the variable-order and distributed-order fractional operators. In Ref. [39], Wang and Zheng discussed the wellposedness of a variable-order time-fraction
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