-robust -norm error estimate of nonuniform Alikhanov scheme for fractional sub-diffusion equation
Introduction
In this work, we consider the following fractional sub-diffusion equation where , are given continuous functions, is the diffusion coefficient. In the following paper, we assume for simplicity of the analysis. The Caputo time derivative with is defined by
The well-known scheme to approximate the Caputo fractional derivative of order is L1 scheme, which has been widely investigated by many researchers, see the recent survey paper [1]. But the L1 scheme can at most get order convergence, so in this paper we investigate a higher order scheme named - scheme which was first proposed by Alikhanov [2] on uniform mesh. Later the - scheme on general meshes was investigated by [3], [4]. Ren et al. [5] established the sharp -norm error estimates for fractional sub-diffusion problems, but their final result may blowup when approximates , so the main goal of this short paper is to present an -robust -norm error estimate for the problem (1), i.e. the final error bound does not blow up when . More precisely, in this paper, we have presented an improved discrete fractional Grönwall inequality of [5] in the linear case, and by employing this inequality an -robust error bound is obtained.
The rest of this paper is organized as follows. In Section 2, we firstly introduce the Alikhanov scheme on graded mesh, then the fully discrete scheme for problem (1) is formulated. The -robust -norm error estimate is given in Section 3. Finally, some numerical results are presented in Section 4.
Notation Throughout this paper, we denote by a generic constant which is independent of the mesh sizes, and the value of may vary from line to line. We call the constant
-robust if it does not blow up when .
Section snippets
Alikhanov scheme on graded mesh
In this paper, we use graded mesh to tackle the initial singularity. Set with for , where is a positive integer, is the grading parameter, particularly represents the uniform grid. Set time step for , , is mesh ratio. For spatial mesh, choose as positive integers, , . Let , , and . We denote by or as nodal
-norm error estimate
For any mesh functions and that vanish on the boundary , we introduce the discrete inner product , norm and semi-norm .
We introduce an improved discrete fractional Grönwall inequality [6] which will be used for our -robust error analysis.
Lemma 3.1 Assume hold and . Let and be given non-negative sequences. For any non-negative sequence , suppose that then for
Numerical experiments
Example 4.1 In (1a) take , , and . We choose which has typical weak singularity at the initial time . The force term .
Table 1, Table 2, Table 3 present errors and orders of convergence for Example 4.1 with different values of and when . From those tables, we could get the temporal convergence order is , which agrees precisely with the theoretical rate of convergence in Theorem 3.5.
Acknowledgment
The research is supported in part by the National Natural Science Foundation of China under Grants No. 11801026, No. 11971482, sponsored by OUC Scientific Research Starting Fund of Introduced Talent, China .
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