α-robust H1-norm error estimate of nonuniform Alikhanov scheme for fractional sub-diffusion equation

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Abstract

The sharp H1-norm error estimate of a finite difference method for two-dimensional time-fractional diffusion equation is established, where the Caputo time-fractional derivative term is approximated by Alikhanov scheme on graded mesh. Under reasonable assumption that the exact solution has typical weak singularity at initial time, the temporal convergence order of the fully scheme is O(Nmin{rα,2}) and the error bound does not blow up when α approximates 1. The theoretical analysis utilizes an improved discrete fractional Grönwall inequality, and the correctness of the theoretical result is verified by numerical experiments.

Introduction

In this work, we consider the following fractional sub-diffusion equation Dtαu(x,y,t)νΔu(x,y,t)=f(x,y,t), for (x,y,t)Ω×(0,T],u(x,y,0)=ϕ(x,y), for (x,y)Ω,u(x,y,t)=μ(x,y,t), for (x,y)Ω,t(0,T], where Ω=(0,l)2, f,ϕ,μ are given continuous functions, ν>0 is the diffusion coefficient. In the following paper, we assume ν=1 for simplicity of the analysis. The Caputo time derivative Dtαu with 0<α<1 is defined by Dtαu(,t)1Γ(1α)s=0t(ts)αsu(,s)ds for (x,y,t)Ω×(0,T].

The well-known scheme to approximate the Caputo fractional derivative of order 0<α<1 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 2α order convergence, so in this paper we investigate a higher order scheme named L2-1σ scheme which was first proposed by Alikhanov [2] on uniform mesh. Later the L2-1σ scheme on general meshes was investigated by [3], [4]. Ren et al. [5] established the sharp H1-norm error estimates for fractional sub-diffusion problems, but their final result may blowup when α approximates 1, so the main goal of this short paper is to present an α-robust H1-norm error estimate for the problem (1), i.e. the final error bound does not blow up when α1. 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 α-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 α-robust H1-norm error estimate is given in Section 3. Finally, some numerical results are presented in Section 4.

Notation

Throughout this paper, we denote by C a generic constant which is independent of the mesh sizes, and the value of C may vary from line to line. We call the constant C α-robust if it does not blow up when α1.

Section snippets

Alikhanov scheme on graded mesh

In this paper, we use graded mesh to tackle the initial singularity. Set 0=t0<t1<t2<<tN=T with tn=T(n/N)r for 0nN, where N is a positive integer, r1 is the grading parameter, particularly r=1 represents the uniform grid. Set time step τn=tntn1 for n=1,2,,N, τ=max1nNτn, ρn=τn/τn+1 is mesh ratio. For spatial mesh, choose M1,M2 as positive integers, h1=l/M1,h2=l/M2, xi=ih1,0iM1,yj=jh2,0jh2. Let Ω¯h={(xi,yj)|0iM1,0jM2}, Ωh=Ω¯hΩ, and Ωh=Ω¯hΩ. We denote by uijn or uhn as nodal

H1-norm error estimate

For any mesh functions v and w that vanish on the boundary Ωh, we introduce the discrete inner product (v,w)=h1h2(x,y)Ωhv(x,y)w(x,y), L2 norm v=(v,v) and H1 semi-norm |v|1=(v,Δhv)=(hv,hv).

We introduce an improved discrete fractional Grönwall inequality [6] which will be used for our α-robust error analysis.

Lemma 3.1

Assume A1A3 hold and 0σ<1/2. Let (ξn)n=1N and (ηn)n=1N be given non-negative sequences. For any non-negative sequence (vn)n=0N, suppose that Dταv2nσvnσξn+(ηn)2 for 1nN,then for

Numerical experiments

Example 4.1

In (1a) take ν=1, l=π, and T=1. We choose u(x,y,t)=(1+tα)sinxsiny which has typical weak singularity at the initial time t=0. The force term f(x,y,t)=Γ(1+α)sinxsiny+2(1+tα)sinxsiny.

Table 1, Table 2, Table 3 present errors and orders of convergence for Example 4.1 with different values of α and M1,M2,N when r=1/α,r=2/α,r=3αα. From those tables, we could get the temporal convergence order is O(Nmin{2,rα}), 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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