H^1-stability of an L2-type method on general nonuniform meshes for subdiffusion equation

by   Chaoyu Quan, et al.

In this work the H^1-stability of an L2 method on general nonuniform meshes is established for the subdiffusion equation. Under some mild constraints on the time step ratio ρ_k, for example 0.4573328≤ρ_k≤ 3.5615528 for all k≥ 2, a crucial bilinear form associated with the L2 fractional-derivative operator is proved to be positive semidefinite. As a consequence, the H^1-stability of L2 schemes can be derived for the subdiffusion equation. In the special case of graded mesh, such positive semidefiniteness holds when the grading parameter 1<r≤ 3.2016538 and therefore the H^1-stability of L2 schemes holds. To the best of our knowledge, this is the first work on the H^1-stability of L2 method on general nonuniform meshes for subdiffusion equation.


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1. Introduction

The time-fractional diffusion equation was derived from continuous time random walks [12, 3], where a fractional derivative in time is introduced to model the memory effect in diffusing materials.

In the past decade, many numerical methods have been proposed to solve the time-fractional diffusion equation. Some of these methods are on the uniform time meshes. For example, the L1 scheme of -order has been well-developed by Langlands and Henry [8], Sun-Wu [17], and Lin-Xu [10], etc. Alikhanov [1] proposed the L2-1 scheme that has second order accuracy in time. An L2 method of -order on uniform meshes is studied in [2] by Gao-Sun-Zhang. In [11], a slightly different L2 fractional-derivative operator is analyzed by Lv-Xu for uniform meshes, where the optimal convergence -order in time is obtained under strong regularity assumptions on the exact solution.

Recently, those methods on nonuniform time meshes for time-fractional diffusion equation have attracted more and more attention, in particular, on the graded meshes. In fact, the exact solution to the time-fractional diffusion equation could have low regularity in general near the initial time , which would deteriorate the convergence rate of the numerical solutions. This motivates researchers to consider nonuniform time meshes to obtain the desired -order convergence rate under low regularity assumptions on the exact solution. For example, Stynes-Riordan-Gracia [16] prove the sharp error analysis of L1 scheme on graded meshes. Kopteva provides a different framework for the analysis of the L1 scheme on graded meshes in two and three spatial dimensions in [6], and analyzes an L2 scheme of -order on graded meshes in [7]. Note that in these works, the -stability of numerical solutions is established.

In addition to the L1 and L2 methods on graded meshes, we mention the convolution quadrature methods with corrections which can also overcome the convergence rate problem for time-fractional diffusion equation, see for example [4, 5] and the references therein.

In this work, we consider the -stability of an L2 method (the same as [7]) on general nonuniform meshes for subdiffusion equation. Note that in general the time steps could change freely, unlike the graded meshes controlled by the grading parameter. For the L2 fractional-derivative operator denoted by , we prove that the following bilinear form


is positive semidefinite under mild restrictions on the time step ratio with ( is the th time step), see Theorem 3.2 for details. In particular, if


such positive semidefiniteness holds. As a consequence, the -stability of the L2 scheme for the subdiffusion equation with homogeneous Dirichlet boundary condition can be derived for all time, see Theorem 4.1. Moreover, in the special case of graded meshes, we show that if the grading parameter


then is positive semidefinite and consequently the -stability of L2 scheme is established.

This work is organized as follows. In Section 2, the derivation, explicit expression and reformulation of L2 fractional-derivative operator are provided. In Section 3, we prove the positive semidefiniteness of the bilinear form under some mild restrictions on the time step ratios. In Section 4, we establish the -stability of the L2 scheme for the subdiffusion equation, based on the positive semidefiniteness result. In Section 5, the special case of graded meshes is discussed.

2. Discrete fractional-derivative operator

In this part we show the derivation, explicit expression and reformulation of L2 operator on an arbitrary nonuniform mesh.

We consider the L2 approximation of the fractional-derivative operator defined by


Take a nonuniform time mesh with . When

, we use the standard linear Lagrangian polynomial interpolating



When , for , we use the standard quadratic Lagrangian polynomial interpolating :


while for , we use the quadratic Lagrangian polynomial defined in (2.3).

Let . At , the fractional derivative is approximated by the discrete fractional-derivative operator


where for ,




It can be verified that , , for , and , , . Furthermore,


always holds for .

Specifically speaking, we can figure out the explicit expressions of and as follows (note that ): for ,


while for ,


We reformulate the discrete fractional derivative in (2.4) as




To establish the -stability of L2-type method for fractional-order parabolic problem, we want to prove the positive semidefiniteness of the following bilinear form


where is the standard inner product over and are two functions defined on and . That is to say, we want to prove that for any defined on and ,


3. Positive semidefiniteness of bilinear form

3.1. Properties of L2 coefficients

We propose the following properties of the L2 coefficients , and in (2.12), which will be useful to establish the positive semidefiniteness of bilinear form .

Lemma 3.1 (Properties of , and ).

For the L2 coefficients given in (2.5) and (2.13), given a nonuniform mesh , the following properties hold:

  • ;

  • ;

  • ;

  • ;

  • ;

  • ;

  • ;

  • .

Furthermore, if the nonuniform mesh , with satisfies


then the following properties of hold:

  • ;

  • .


We first provide two equivalent forms of in (2.5) as follows:




It is easy to see , i.e., (P1) holds.

Combining (3.2) and (3.3), we have


where we use the form (3.2) for and the form (3.3) for . Therefore (P3) holds. Moreover, for any fixed ,


decrease w.r.t. . As a consequence, (3.3) and (3.4) result in


i.e., the properties (P2) and (P4) hold.

We now turn to prove the properties of and . For in (2.5), we have


This is the property (P5). Since , we have for and the property (P7) holds. For any fixed ,


decreases w.r.t. , implying that


i.e., the property (P6). Combining this with property (P2), the property (P8) holds.

We now prove the property (P9). Combining (3.4) and (3.7) gives


Note that for any fixed ,


decreases w.r.t. , and


which imply




Using (3.14) and the fact


we can derive from (3.10) that


To ensure the property (P9), we need


that is the condition (3.1).

We now prove the last property (P10). From (3.10), we can derive that


The convexity of the function gives


and for fixed , it is easy to see that


decreases w.r.t. . Then we can get the following result similar to (3.16):