Mathematical Problems in Engineering

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Coupled Numerical Methods in Engineering Analysis

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Volume 2011 |Article ID 171620 | https://doi.org/10.1155/2011/171620

Yunying Zheng, Zhengang Zhao, "A Fully Discrete Galerkin Method for a Nonlinear Space-Fractional Diffusion Equation", Mathematical Problems in Engineering, vol. 2011, Article ID 171620, 20 pages, 2011. https://doi.org/10.1155/2011/171620

A Fully Discrete Galerkin Method for a Nonlinear Space-Fractional Diffusion Equation

Academic Editor: Delfim Soares
Received13 May 2011
Revised27 Jul 2011
Accepted27 Jul 2011
Published05 Oct 2011

Abstract

The spatial transport process in fractal media is generally anomalous. The space-fractional advection-diffusion equation can be used to characterize such a process. In this paper, a fully discrete scheme is given for a type of nonlinear space-fractional anomalous advection-diffusion equation. In the spatial direction, we use the finite element method, and in the temporal direction, we use the modified Crank-Nicolson approximation. Here the fractional derivative indicates the Caputo derivative. The error estimate for the fully discrete scheme is derived. And the numerical examples are also included which are in line with the theoretical analysis.

1. Introduction

The normal diffusive motion is modeled to describe the standard Brownian motion. The relation between the flow and the divergence of the particle displacement represents 𝐽(π‘₯,𝑑)=βˆ’π‘Žπœ•π‘πœ•π‘₯+𝑏𝑐,(1.1) where 𝐽 is the diffusive flow. Inserting the above equation into the equation of mass conservation πœ•π½πœ•π‘₯=βˆ’πœ•π‘πœ•π‘‘,(1.2) we obtain the standard convection-diffusion equation. From the viewpoint of physics, it means that during the method of time random walkers, the overall particle displacement up to time 𝑑 can be represented as a sum of independent random steps, in the case that both the mean-squared displacement per step and the mean time needed to perform a step are finite. The measured variance growth in the direction of flow of tracer plumes is typically at a Fickian rate, ⟨(π‘βˆ’π‘)2βŸ©βˆΌπ‘‘.

The transport process in fractal media cannot be described with the normal diffusion. The process is nonlocal and it does not follow the classical Fickian law. It depicts a particle in spreading tracer cloud which has a standard deviation, and which grows like 𝑑2𝛼 for some 0<𝛼<1, excluding the Fickian case 𝛼=1/2. The description of anomalous diffusion means that the measure variance growth in the direction of flow has a deviation from the Fickian case, it follows the super-Fickian rate ⟨(π‘βˆ’π‘)2βŸ©βˆΌπ‘‘2𝛼 when 𝛼>1/2, or does the subdiffusion rate ⟨(π‘βˆ’π‘)2βŸ©βˆΌπ‘‘2𝛼 if 0<𝛼<1/2. With the help of the continuous time random walk and the Fourier transform, the governing equation with space fractional derivative can be derived as follows πœ•π‘’ξ‚€πœ•π‘‘=π·π‘Ž(𝑒)π‘Žπ·π›½π‘₯𝑒+𝑏(𝑒)𝐷𝑒+𝑓(π‘₯,𝑑,𝑒),0<𝛽<1,(1.3) where 𝐷 denotes integer derivative respect to π‘₯, and 𝐷𝛽 is fractional derivative. There are some authors studying the spacial anomalous diffusion equation in theoretical analysis and numerical simulations [1–10]. Now the fractional anomalous diffusion becomes a hot topic because of its widely applications in the evolution of various dynamical systems under the influence of stochastic forces. For example, it is a well-suited tool for the description of anomalous transport processes in both absence and presence of external velocities or force fields. Since the groundwater velocities span many orders of magnitude and give rise to diffusion-like dispersion (a term that combines molecular diffusion and hydrodynamic dispersion), the fractional diffusion is an important process in hydrogeology. It can be used to describe the systems with reactions and diffusions across a wide range of applications including nerve cell signaling, animal coat patterns, population dispersal, and chemical waves. In general, fractional anomalous diffusions have numerous applications in statistical physics, biophysics, chemistry, hydrogeology, and biology [4, 11–20].

In this paper, we mainly study one kind of typical nonlinear space-fractional partial differential equations by using the finite element method, which reads in the following form: πœ•π‘’ξ‚€πœ•π‘‘=π·π‘Ž(𝑒)π‘Žπ·π›½π‘₯𝑒],+𝑏(𝑒)𝐷𝑒+𝑓(π‘₯,𝑑,𝑒),π‘₯∈Ω,π‘‘βˆˆ(0,π‘‡π‘’βˆ£π‘‘=0=πœ‘(π‘₯),π‘₯∈Ω,π‘’βˆ£πœ•Ξ©],=𝑔,π‘‘βˆˆ(0,𝑇(1.4) where Ξ© is a spacial domain with boundary πœ•Ξ©, 𝐷𝛽 is the 𝛽th (0<𝛽<1) order fractional derivative with respect to the space variable π‘₯ in the Caputo sense (which will be introduced later on),π‘Ž,𝑏,𝑓 are functions ofπ‘₯,𝑑,𝑒,πœ‘ and 𝑔 are known functions which satisfy the conditions requested by the theorem of error estimations.

The rest of this paper is constructed as follows. In Section 2 the fractional integral, fractional derivative, and the fractional derivative spaces are introduced. The error estimates of the finite element approximation for (1.4) are studied in Section 3, and in Section 4, numerical examples are taken to verify the theoretical results derived in Section 3.

2. Fractional Derivative Space

In this section, we firstly introduce the fractional integral (or Riemann-Liouville integral), the Caputo fractional derivative, and their corresponding fractional derivative space.

Definition 2.1. The 𝛼th order left and right Riemann-Liouville integrals of function 𝑒(π‘₯) are defined as follows π‘ŽπΌπ›Όπ‘₯1𝑒(π‘₯)=ξ€œΞ“(𝛼)π‘₯π‘Ž(π‘₯βˆ’π‘ )π›Όβˆ’1𝑒(𝑠)𝑑𝑠,π‘₯𝐼𝛼𝑏1𝑒(π‘₯)=Ξ“ξ€œ(𝛼)𝑏π‘₯(π‘ βˆ’π‘₯)π›Όβˆ’1𝑒(𝑠)𝑑𝑠,(2.1) where 𝛼>0, and Ξ“(β‹…) is the Gamma function.

Definition 2.2. The 𝛼th order Caputo derivative of function 𝑒(π‘₯) is defined as, π‘Žπ·π›Όπ‘₯𝑒(π‘₯)=π‘ŽπΌπ‘₯π‘›βˆ’π›Όπ‘‘π‘›π‘’(π‘₯)𝑑π‘₯𝑛,π‘›βˆ’1<𝛼<π‘›βˆˆπ‘+,π‘₯𝐷𝛼𝑏𝑒(π‘₯)=(βˆ’1)𝑛π‘₯πΌπ‘π‘›βˆ’π›Όπ‘‘π‘›π‘’(π‘₯)𝑑π‘₯𝑛,π‘›βˆ’1<𝛼<π‘›βˆˆπ‘+.(2.2) The 𝛼th order Riemann-Liouville derivative of function 𝑒(π‘₯) is defined by changing the order of integration and differentiation.

Lemma 2.3 (see [8]). If 𝑒(0)=𝑒′(0)=β‹―=𝑒(π‘›βˆ’1)(0)=0, then the Caputo fractional derivative is equal to the Riemann-Liouville derivative.

Definition 2.4. The fractional derivative space 𝐽𝛼(Ξ©) is defined as follows: 𝐽𝛼(Ξ©)=π‘’βˆˆπΏ2(Ξ©)βˆΆπ‘Žπ·π›Όπ‘₯π‘’βˆˆπΏ2ξ€Ύ(Ξ©),π‘›βˆ’1≀𝛼<𝑛,(2.3) endowed with the seminorm |𝑒|𝐽𝛼=β€–β€–π‘Žπ·π›Όπ‘₯𝑒‖‖𝐿2(Ξ©),(2.4) and the norm ‖𝑒‖𝐽𝛼=|𝑒|2𝐽𝛼+[𝛼]π‘˜β‰€β€–β€–π·π‘˜π‘’β€–β€–2ξƒͺ1/2.(2.5)

Let 𝐽𝛼0(Ξ©) denote the closure of 𝐢∞0(Ξ©) with respect to the above norm and seminorm.

Definition 2.5. Define the seminorm |𝑒|𝐻𝛼=β€–|𝑖𝑀|𝛼𝐹‖(𝑒)𝐿2(Ξ©),(2.6) and the norm ‖𝑒‖𝐻𝛼=|𝑒|2𝐻𝛼+[𝛼]π‘˜β‰€β€–β€–π·π‘˜π‘’β€–β€–2ξƒͺ1/2,(2.7) where 𝑖 is the imaginary unit, and 𝐹 is the Fourier transform, and which can define another fractional derivative space 𝐻𝛼(Ξ©).

Let 𝐻𝛼0(Ξ©) denote the closure of 𝐢∞0(Ξ©) with respect to the norm and seminorm.

Definition 2.6. The fractional space 𝐽𝛼𝑠(Ξ©) is defined below 𝐽𝛼𝑠(Ξ©)=π‘’βˆˆπΏ2(Ξ©)βˆΆπ‘Žπ·π›Όπ‘₯π‘’βˆˆπΏ2(Ξ©),π‘₯π·π›Όπ‘π‘’βˆˆπΏ2ξ€Ύ(Ξ©),π‘›βˆ’1≀𝛼<𝑛,(2.8) endowed with the seminorm |𝑒|𝐽𝛼𝑠=|||ξ€·π‘Žπ·π›Όπ‘₯𝑒,π‘₯𝐷𝛼𝑏𝑒1/2|||𝐿2(Ξ©),(2.9) and the norm ‖𝑒‖𝐽𝛼𝑠=[𝛼]π‘˜β‰€β€–β€–π·π‘˜π‘’β€–β€–2+|𝑒|2𝐽𝛼𝑠ξƒͺ1/2.(2.10)

Theorem 2.7 (see [3, 6]). 𝐽𝛼𝑠, 𝐽𝛼, and 𝐻𝛼 are equal with equivalent seminorm and norm.
The following are some useful results.

Lemma 2.8 (see [3]). For π‘’βˆˆπ½π›Ό0(Ξ©), 0<𝛽<𝛼, then π‘Žπ·π›Όπ‘₯𝑒(π‘₯)=π‘Žπ·π‘₯π‘Žπ›Όβˆ’π›½π·π›½π‘₯𝑒.(2.11)

Lemma 2.9 (see [2]). For π‘’βˆˆπ»π›Ό0(Ξ©), one has ‖𝑒‖𝐿2(Ξ©)≀𝑐|𝑒|𝐻𝛼0.(2.12) For 0<𝛽<𝛼, |𝑒|𝐻𝛽0(Ξ©)≀𝑐|𝑒|𝐻𝛼0.(2.13)

Since 𝐽𝛼𝑠, 𝐽𝛼, and 𝐻𝛼 are equal with equivalent seminorm and norm, the norms with each space which will be used following are without distinction, and the notations are used seminorm |β‹…|𝛼 and norm ‖⋅‖𝛼.

3. Finite Element Approximation

Let Ξ©=[π‘Ž,𝑏], and 0≀𝛽<1. Define 𝛼=(1+𝛽)/2. In this section, we will formulate a fully discrete Galerkin finite element method for a type of nonlinear anomalous diffusion equation as follows.

Problem 1 (Nonlinear spacial anomalous diffusion equation). We consider equations of the form πœ•π‘’ξ‚€πœ•π‘‘=π·π‘Ž(𝑒)π‘Žπ·π›½π‘₯𝑒],],+𝑏(𝑒)𝐷𝑒+𝑓(π‘₯,𝑑,𝑒),(π‘₯,𝑑)βˆˆΞ©Γ—(0,𝑇𝑒(π‘₯,𝑑)=πœ™(π‘₯,𝑑),π‘₯βˆˆπœ•Ξ©Γ—(0,𝑇𝑒(π‘₯,0)=𝑔(π‘₯),π‘₯∈Ω.(3.1) We always assume that 0<π‘š<π‘Ž(𝑒)<𝑀,0<π‘š<𝑏(𝑒)<𝑀,0<π‘š<𝑓(𝑒)<𝑀.(3.2)

The algorithm and analysis in this paper are applicable for a large class of linear and nonlinear functions (including polynomials and exponentials) in the unknown variables. Throughout the paper, we assume the following mild Lipschitz continuity conditions onβ€‰β€‰π‘Ž,𝑏,  and  𝑓: there exist positive constants 𝐿 and 𝑐 such that for π‘₯∈Ω,π‘‘βˆˆ(0,𝑇], and 𝑠,π‘‘βˆˆπ‘…, ||||π‘Ž(π‘₯,𝑑,𝑠)βˆ’π‘Ž(π‘₯,𝑑,π‘Ÿ)≀𝐿|π‘ βˆ’π‘Ÿ|,(3.3)||||𝑏(π‘₯,𝑑,𝑠)βˆ’π‘(π‘₯,𝑑,π‘Ÿ)≀𝐿|π‘ βˆ’π‘Ÿ|,(3.4)||||𝑓(π‘₯,𝑑,𝑠)βˆ’π‘“(π‘₯,𝑑,π‘Ÿ)≀𝐿|π‘ βˆ’π‘Ÿ|.(3.5)

In order to derive a variational form of Problem 1, we suppose that 𝑒 is a sufficiently smooth solution of Problem 1. Multiplying an arbitrary π‘£βˆˆπ»π›Ό0(Ξ©) in both sides yields ξ€œΞ©πœ•π‘’ξ€œπœ•π‘‘π‘£π‘‘π‘₯=Ξ©π·ξ‚€π‘Ž(𝑒)π‘Žπ·π›½π‘₯π‘’ξ‚ξ€œπ‘£π‘‘π‘₯+Ξ©ξ€œπ‘(𝑒)𝐷𝑒𝑣𝑑π‘₯+Ω𝑓(π‘₯,𝑑,𝑒)𝑣𝑑π‘₯.(3.6)

Rewriting the above expression yields ξ€œΞ©πœ•π‘’ξ€œπœ•π‘‘π‘£π‘‘π‘₯+Ξ©π‘Ž(𝑒)π‘Žπ·π›½π‘₯ξ€œπ‘’π·π‘£π‘‘π‘₯βˆ’Ξ©ξ€œπ‘(𝑒)𝐷𝑒𝑣𝑑π‘₯=Ω𝑓(π‘₯,𝑑,𝑒)𝑣𝑑π‘₯.(3.7)

We define the associated bilinear form π΄βˆΆπ½π›Ό0(Ξ©)×𝐽𝛼0(Ξ©)→𝑅 as 𝐴(𝑒,𝑣)=π‘Ž(𝑒)π‘Žπ·π›½π‘₯𝑒,π·π‘£βˆ’(𝑏(𝑒)𝐷𝑒,𝑣),(3.8) where (β‹…,β‹…) denotes the inner product on 𝐿2(Ξ©) and 𝐽𝛼0(Ξ©).

For given π‘“βˆˆπ½βˆ’π›Ό(Ξ©), we define the associated function πΉβˆΆπ½π›Ό0(Ξ©)→𝑅 as 𝐹(𝑣)=βŸ¨π‘“,π‘£βŸ©.(3.9)

Definition 3.1. A function π‘’βˆˆπ½π›Ό0(Ξ©) is a variational solution of Problem 1 provided that ξ‚€πœ•π‘’ξ‚πœ•π‘‘,𝑣+𝐴(𝑒,𝑣)=𝐹(𝑣),βˆ€π‘£βˆˆπ½π›Ό0(Ξ©).(3.10)

Now we are ready to describe a fully discrete Galerkin finite element method to solve nonlinear Problem 1. In our new scheme, the finite element trial and test spaces for Problem 1 are chosen to be same.

For a positive integer 𝑁, let βˆπ‘‘={𝑑𝑛}𝑁𝑛=0 be a uniform partition of the time interval (0,𝑇] such that 𝑑𝑛=π‘›πœ, where 𝜏=𝑇/𝑁, and let π‘‘π‘›βˆ’1/2=π‘‘π‘›βˆ’πœ/2. Throughout the paper, we use the following notation for a function πœ™: πœ™π‘›ξ€·π‘‘=πœ™π‘›ξ€Έ,πœ•π‘‘πœ™π‘›=πœ™π‘›βˆ’πœ™π‘›βˆ’1𝜏,πœ™π‘›=πœ™π‘›+πœ™π‘›βˆ’12,ξ‚πœ™π‘›=3πœ™π‘›βˆ’1βˆ’πœ™π‘›βˆ’22.(3.11)

Let π’¦β„Ž={𝐾} be a partition of spatial domain Ξ©. Define β„Žπ‘˜ as the diameter of the element 𝐾 and β„Ž=maxπΎβˆˆπ’¦β„Žβ„ŽπΎ. And let π‘†β„Ž be a finite element space π‘†β„Ž=ξ€½π‘£βˆˆπ»π›Ό0(Ξ©)βˆΆπ‘£βˆ£πΎβˆˆπ‘ƒπ‘Ÿβˆ’1(𝐾),πΎβˆˆπ’¦β„Žξ€Ύ,(3.12) where π‘ƒπ‘Ÿβˆ’1(𝐾) is the set of polynomials of degree π‘Ÿβˆ’1 on a given domain 𝐾. And the functions in π‘†β„Ž are continuous on Ξ©. Our fully discrete quadrature scheme to solve Problem 1 is to find π‘’β„Ž: for π‘£βˆˆπ‘†β„Ž such that ξ‚€πœ•π‘‘π‘’π‘›β„Žξ‚+ξ‚€π‘Žξ€·,π‘£Μƒπ‘’π‘›β„Žξ€Έπ‘Žπ·π›½π‘₯π‘’π‘›β„Žξ‚βˆ’ξ€·π‘ξ€·,π·π‘£Μƒπ‘’π‘›β„Žξ€Έπ·π‘’π‘›β„Žξ€Έ=𝑓,π‘£Μƒπ‘’π‘›β„Žξ€Έξ¬,𝑣.(3.13)

The linear systems in the above equation requires selecting the value of 𝑒0β„Ž and 𝑒1β„Ž. Given 𝑒0β„Ž depending on the initial data 𝑔(π‘₯), we select 𝑒1β„Ž by solving the following predictor-corrector linear systems: ξƒ©π‘’β„Ž1,0βˆ’π‘’0β„Žπœξƒͺ+ξƒ©π‘Žξ€·π‘’,𝑣0β„Žξ€Έπ‘Žπ·π›½π‘₯π‘’β„Ž1,0+𝑒0β„Ž2ξƒͺβˆ’ξƒ©π‘ξ€·π‘’,𝐷𝑣0β„Žξ€Έπ·π‘’β„Ž1,0+𝑒0β„Ž2ξƒͺ=𝑓𝑒,𝑣0β„Žξ€Έξ¬,𝑒,𝑣1β„Žβˆ’π‘’0β„Žπœξƒͺ+ξƒ©π‘Žξƒ©π‘’,π‘£β„Ž1,0+𝑒0β„Ž2ξƒͺπ‘Žπ·π›½π‘₯𝑒1β„Ž+𝑒0β„Ž2ξƒͺβˆ’ξƒ©π‘ξƒ©π‘’,π·π‘£β„Ž1,0+𝑒0β„Ž2ξƒͺ𝐷𝑒1β„Ž+𝑒0β„Ž2ξƒͺ=𝑓𝑒,π‘£β„Ž1,0+𝑒0β„Ž2ξƒͺξ„•.,𝑣(3.14)

Lemma 3.2. For 𝑒,𝑣,π‘€βˆˆπ½π›Όπ‘ ,0(Ξ©),0<π‘šβ‰€π‘Ž(𝑒)≀𝑀,𝛼=(1+𝛽)/2, there exist constants 𝛾1,𝛾2 such that ξ‚€π‘Ž(𝑒)π‘Žπ·π›½π‘₯𝑒,𝐷𝑣≀𝛾1‖𝑒‖𝛼⋅‖𝑣‖𝛼,ξ‚€π‘Ž(𝑀)π‘Žπ·π›½π‘₯𝑣,𝐷𝑣β‰₯𝛾2‖𝑣‖2𝛼.(3.15)

Proof. With the assumption of π‘Ž(𝑒) in (3.3) and the property of dual space ξ‚€π‘Ž(𝑀)π‘Žπ·π›½π‘₯≀‖‖𝑒,π·π‘£π‘Ž(𝑀)π‘Žπ·π›½π‘₯𝑒‖‖1βˆ’π›Όβ‹…β€–π·π‘£β€–βˆ’(1βˆ’π›Ό)≀𝑀𝑐‖𝑒‖1βˆ’π›Ό+π›½β‹…β€–π‘£β€–βˆ’(1βˆ’π›Ό)+1≀𝛾1‖𝑒‖𝛼⋅‖𝑣‖𝛼,ξ‚€π‘Ž(𝑀)π‘Žπ·π›½π‘₯𝑣,𝐷𝑣=βˆ’π·π‘Ž(𝑀)π‘Žπ·π›½π‘₯𝑣,𝑣=βˆ’π‘Žπ·π‘₯(1βˆ’π›½)/2π‘Ž(𝑀)π‘Žπ·π›½π‘₯𝑣,π‘₯𝐷𝑏(1+𝛽)/2𝑣β‰₯π‘š|𝑣|2𝐽𝛼𝑠β‰₯𝛾2‖𝑣‖2𝛼.(3.16)

Lemma 3.3 (see [2]). For Ξ©βŠ‚π‘…π‘›,𝛼>𝑛/4,𝑣,π‘€βˆˆπ»π›Ό0(Ξ©),πœ€>0, one has (𝑣𝑏(𝑀),βˆ‡π‘£)≀𝑐0(π‘žπœ€)βˆ’π‘/π‘žπ‘β€–β€–βˆ‡π‘(𝑀)𝑝⋅‖𝑣‖2+πœ€β€–π‘£β€–2𝛼,(3.17) where 𝑝=4𝛼/(4π›Όβˆ’π‘›),π‘ž=4𝛼/𝑛.

Theorem 3.4. Let π‘’π‘›β„Ž be bounded, then for a sufficiently small step 𝜏, there exists a unique solution π‘’π‘›β„Žβˆˆπ‘†β„Ž satisfying scheme (3.13).

Proof. As scheme represents a finite system of problem, the continuity and coercivity of (π‘’π‘›β„Ž,πœ”π‘›β„Ž)/𝜏+𝐴(π‘’π‘›β„Ž,πœ”π‘›β„Ž) is the sufficient and essential condition for the existence and uniqueness of π‘’π‘›β„Ž. Let 𝑣=π‘’π‘›β„Ž,𝑀=πœ”π‘›β„Ž, then (𝑣,𝑣)𝜏+𝐴(𝑣,𝑣)=(𝑣,𝑣)𝜏+ξ‚€π‘Ž(𝑀)π‘Žπ·π›½π‘₯β‰₯𝑣,π·π‘£βˆ’(𝑏(𝑀)𝐷𝑣,𝑣)‖𝑣‖2𝜏+𝛾2β€–π‘£β€–π›Όβˆ’π‘0‖𝐷𝑏(𝑀)β€–2‖𝑣‖2βˆ’πœ€β€–π‘£β€–2𝛼=𝛾2ξ€Έβˆ’πœ€β€–π‘£β€–2𝛼+ξ€·πœβˆ’1βˆ’π‘0‖‖𝐷𝑏(𝑀)2‖𝑣‖2β‰₯𝑐‖𝑣‖2𝛼.(3.18) For the chosen sufficiently small 𝜏, the above inequality holds. (𝑣,𝑀)𝜏+𝐴(𝑣,𝑀)=(𝑣,𝑀)𝜏+ξ‚€π‘Ž(𝑒)π‘Žπ·π›½π‘₯≀𝑣,𝐷𝑀+(𝐷𝑏(𝑒)𝑣,𝐷𝑀)β€–π‘’β€–β‹…β€–π‘€β€–πœ+𝛾1‖𝑣‖𝛼‖𝑀‖𝛼+‖≀‖𝑣‖⋅‖𝐷(𝑏(𝑒)𝑀)β€–π‘’β€–β‹…β€–π‘€β€–πœ+𝛾1‖𝑣‖𝛼‖𝑀‖𝛼+π‘€β€–π‘£β€–β‹…β€–π‘€β€–β„Žβ‰€π‘β€–π‘£β€–π›Όβ€–π‘€β€–π›Ό.(3.19) Hence, the scheme (3.13) is uniquely solvable for π‘’π‘›β„Ž.
Let πœŒπ‘›=π‘ƒβ„Žπ‘’π‘›βˆ’π‘’π‘›, and πœƒπ‘›=π‘’π‘›β„Žβˆ’π‘ƒβ„Žπ‘’π‘›, then π‘’π‘›β„Žβˆ’π‘’π‘›=π‘’π‘›β„Žβˆ’π‘ƒβ„Žπ‘’π‘›+π‘ƒβ„Žπ‘’π‘›βˆ’π‘’π‘›=πœƒπ‘›+πœŒπ‘›,(3.20) where π‘ƒβ„Žπ‘’π‘› is a Rits-Galerkin projection operator defined as follows: ξ‚€π‘Ž(𝑀)π‘Žπ·π›½π‘₯ξ€·π‘’π‘›βˆ’π‘ƒβ„Žπ‘’π‘›ξ€Έξ‚ξ‚€π‘Žξ€·π‘’,𝐷𝑣=0,0ξ€Έπ‘Žπ·π›½π‘₯ξ€·π‘’π‘›βˆ’π‘ƒβ„Žπ‘’π‘›ξ€Έξ‚,𝐷𝑣=0.(3.21)

Lemma 3.5. Let π‘Ž(𝑒),𝑏(𝑒) be smooth functions on Ξ©, 0<π‘šβ‰€π‘Ž(𝑒),𝑏(𝑒)≀𝑀, and π‘ƒβ„Žπ‘’π‘› is defined as above, then β€–β€–π‘Žπ·π›Όπ‘₯ξ€·π‘’π‘›βˆ’π‘ƒβ„Žπ‘’π‘›ξ€Έβ€–β€–β‰€π‘β„Žπ‘˜+1βˆ’π›Όβ€–π‘’β€–π‘˜+1,β€–ξ€·π‘ƒβ„Žπ‘’π‘›βˆ’π‘’π‘›ξ€Έβ€–β‰€π‘β„Žπ‘˜+1β€–π‘’β€–π‘˜+1.(3.22)

Proof. Using the definition of π‘ƒβ„Žπ‘’π‘›, one gets β€–β€–π‘Žπ·π›Όπ‘₯ξ€·π‘ƒβ„Žπ‘’π‘›βˆ’π‘’π‘›ξ€Έβ€–β€–2=||ξ€·π‘Žπ·π›Όπ‘₯ξ€·π‘ƒβ„Žπ‘’π‘›βˆ’π‘’π‘›ξ€Έ,π‘Žπ·π›Όπ‘₯ξ€·π‘ƒβ„Žπ‘’π‘›βˆ’π‘’π‘›||β€–β€–ξ€Έξ€Έβ‰€π‘π‘Žπ·π›Όπ‘₯ξ€·π‘ƒβ„Žπ‘’π‘›βˆ’π‘’π‘›ξ€Έβ€–β€–β‹…β€–β€–π‘Žπ·π›Όπ‘₯(πœ’βˆ’π‘’π‘›)β€–β€–,(3.23) where πœ’βˆˆπ‘†β„Ž. Utilizing the interpolation of πΌβ„Žπ‘’π‘› leads to β€–β€–π‘Žπ·π›Όπ‘₯ξ€·π‘ƒβ„Žπ‘’π‘›βˆ’π‘’π‘›ξ€Έβ€–β€–β‰€infπœ’βˆˆπ‘†β„Žπ‘β€–πœ’βˆ’π‘’β€–π›Όβ€–β€–πΌβ‰€π‘β„Žπ‘’π‘›βˆ’π‘’π‘›β€–β€–π›Όβ‰€π‘β„Žπ‘˜+1βˆ’π›Όβ€–π‘’β€–π‘˜+1.(3.24) Next we estimate β€–π‘ƒβ„Žπ‘’π‘›βˆ’π‘’π‘›β€–. For allπœ™βˆˆπΏ2(Ξ©), 𝑀 is the solution of the following equation: βˆ’π‘Žπ·π‘₯2𝛼𝑀=πœ™,π‘€βˆˆΞ©,𝑀=0,π‘€βˆˆπœ•Ξ©.(3.25)
So we have ‖𝑀‖2𝛼≀𝛾3β€–πœ™β€–.(3.26) For allπœ’βˆˆπ‘†β„Ž, with the help of approximation properties of π‘†β„Ž and the weak form, we can obtain ξ€·π‘ƒβ„Žπ‘’π‘›βˆ’π‘’π‘›ξ€Έξ€·π‘ƒ,πœ™=βˆ’β„Žπ‘’π‘›βˆ’π‘’π‘›,π‘Žπ·π‘₯2𝛼𝑀=βˆ’π‘₯π·π›Όπ‘ξ€·π‘ƒβ„Žπ‘’π‘›βˆ’π‘’π‘›ξ€Έ,π‘Žπ·π›Όπ‘₯𝑀=βˆ’π‘₯π·π›Όπ‘ξ€·π‘ƒβ„Žπ‘’π‘›βˆ’π‘’π‘›ξ€Έ,π‘Žπ·π›Όπ‘₯≀‖‖𝑃(π‘€βˆ’πœ’)β„Žπ‘’π‘›βˆ’π‘’π‘›β€–β€–π›Όβ€–π‘€βˆ’πœ’β€–π›Όβ‰€β€–β€–π‘ƒβ„Žπ‘’π‘›βˆ’π‘’π‘›β€–β€–π›Όinfπœ’βˆˆπ‘†β„Žβ€–π‘€βˆ’πœ’β€–π›Όβ‰€π‘β„Žπ‘Ÿβˆ’π›Όβ€–π‘’β€–π‘Ÿβ„Žπ›Όβ€–π‘€β€–2𝛼=π‘β„Žπ‘Ÿβ€–π‘’β€–π‘Ÿβ€–β€–β€–π‘ƒπœ™β€–,β„Žπ‘’π‘›βˆ’π‘’π‘›β€–β€–=sup0β‰ πœ™βˆˆπΏ2(Ξ©)ξ€·π‘ƒβ„Žπ‘’π‘›βˆ’π‘’π‘›ξ€Έ,πœ™β€–πœ™β€–β‰€π‘β„Žπ‘Ÿβ€–π‘’β€–π‘Ÿ.(3.27)

Lemma 3.6 (see [21]). Let π‘‡β„Ž,0<β„Žβ‰€1, denote a quasiuniform family of subdivisions of a polyhedral domain Ξ©βŠ‚π‘…π‘‘. Let (πΎξ…ž,𝑃,𝑁) be a reference finite element such that π‘ƒβŠ‚π‘Šπ‘™,𝑝(πΎξ…ž)βˆ©π‘Šπ‘š,π‘ž(πΎξ…ž) is a finite-dimensional space of functions on πΎξ…ž,𝑁 is a basis for 𝑃′, where 1β‰€π‘β‰€βˆž,1β‰€π‘β‰€βˆž, and 0β‰€π‘šβ‰€π‘™. For πΎβˆˆπ‘‡β„Ž, let (𝐾,𝑃𝐾,𝑁𝐾) be the affine equivalent element, and π‘‰β„Ž=π‘£βˆΆπ‘£ is measurable and 𝑣|πΎβˆˆπ‘ƒπΎ,forallπΎβˆˆπ‘‡β„Ž. Then there exists a constant 𝐢=𝐢(𝑙,𝑝,π‘ž) such that βŽ‘βŽ’βŽ’βŽ£ξ“π‘˜βˆˆπ‘‡β„Žβ€–π‘£β€–2π‘Šπ‘™,𝑝(𝐾)⎀βŽ₯βŽ₯⎦1/π‘β‰€πΆβ„Žπ‘šβˆ’π‘™+min(0,𝑑/π‘βˆ’π‘‘/π‘ž)β‹…βŽ‘βŽ’βŽ’βŽ£ξ“π‘˜βˆˆπ‘‡β„Žβ€–π‘£β€–π‘žπ‘Šπ‘š,π‘ž(𝐾)⎀βŽ₯βŽ₯⎦1/π‘ž.(3.28) The following Gronwall’s lemma is useful for the error analysis later on.

Lemma 3.7 (see [2]). Let Δ𝑑,𝐻 and π‘Žπ‘›,𝑏𝑛,𝑐𝑛,𝛾𝑛 (for integer 𝑛β‰₯0) be nonnegative numbers such that π‘Žπ‘+Δ𝑑𝑁𝑛=0𝑏𝑛≀Δ𝑑𝑁𝑛=0π›Ύπ‘›π‘Žπ‘›+Δ𝑑𝑁𝑛=0𝑐𝑛+𝐻,(3.29) for 𝑁β‰₯0. Suppose that Δ𝑑𝛾𝑛<1, for all 𝑛, and set πœŽπ‘›=(1βˆ’Ξ”π‘‘π›Ύπ‘›)βˆ’1. Then π‘Žπ‘+Δ𝑑𝑁𝑛=0𝑏𝑛≀expΔ𝑑𝑁𝑛=0πœŽπ‘›π›Ύπ‘›ξƒͺΔ𝑑𝑁𝑛=0𝑐𝑛+𝐻,(3.30) for 𝑁β‰₯0.
The following norms are also used in the analysis: β€–|𝑣|β€–βˆž,π‘˜=max0β‰€π‘›β‰€π‘β€–π‘£π‘›β€–π‘˜,β€–|𝑣|β€–0,π‘˜=𝑁𝑛=0πœβ€–π‘£π‘›β€–2π‘˜ξƒ­1/2.(3.31)

Theorem 3.8. Assume that Problem 1 has a solution 𝑒 satisfying 𝑒𝑑𝑑,π‘’π‘‘π‘‘π‘‘βˆˆπΏ2(0,𝑇,𝐿2(Ξ©)) with 𝑒,π‘’π‘‘βˆˆπΏ2(0,𝑇,π»π‘˜+1). If Ξ”π‘‘β‰€π‘β„Ž, then the finite element approximation is convergent to the solution of Problem 1 on the interval (0,T], as Δ𝑑,β„Žβ†’0. The approximation π‘’β„Ž also satisfies the following error estimates β€–β€–π‘’βˆ’π‘’β„Žβ€–β€–0,π›Όξ€·β„Žβ‰€πΆπ‘˜+1‖‖𝑒𝑑‖‖0,π‘˜+1+β„Žπ‘˜+1βˆ’π›Όβ€–π‘’β€–0,π‘˜+1+𝜏2‖‖𝑒𝑑𝑑‖‖0,0+πœβ„Žπ‘˜+1βˆ’π›Όβ€–β€–π‘’π‘‘π‘‘β€–β€–0,π‘˜+1+𝜏2‖‖𝑒𝑑𝑑𝑑‖‖0,0ξ€Έ,β€–β€–(3.32)π‘’βˆ’π‘’β„Žβ€–β€–βˆž,0ξ€·β„Žβ‰€πΆπ‘˜+1‖‖𝑒𝑑‖‖0,π‘˜+1+β„Žπ‘˜+1βˆ’π›Όβ€–π‘’β€–0,π‘˜+1+𝜏2‖‖𝑒𝑑𝑑𝑑‖‖0,0+πœβ„Žπ‘˜+1βˆ’π›Όβ€–β€–π‘’π‘‘π‘‘β€–β€–0,π‘˜+1+𝜏2‖‖𝑒𝑑𝑑‖‖0,0+β„Žπ‘˜+1‖𝑒‖2∞,π‘˜+1ξ€Έ.(3.33)

Proof. For 𝑑=π‘‘π‘›βˆ’πœ/2=π‘‘π‘›βˆ’1/2,𝑛=0,1,…,𝑁, find π‘’π‘›βˆ’1/2 such that ξ€·πœ•π‘‘π‘’π‘›βˆ’1/2ξ€Έ+ξ€·π‘Žξ€·π‘’,π‘£π‘›βˆ’1/2ξ€Έπ‘Žπ·π‘₯π‘’π‘›βˆ’1/2ξ€Έβˆ’ξ€·π‘ξ€·π‘’,π·π‘£π‘›βˆ’1/2ξ€Έπ·π‘’π‘›βˆ’1/2ξ€Έ=𝑓𝑒,π‘£π‘›βˆ’1/2,𝑣.(3.34)
Subtracting the above equation from the fully discrete scheme (3.13), and substituting π‘’π‘›β„Žβˆ’π‘’π‘›=(π‘’π‘›β„Žβˆ’π‘ƒβ„Žπ‘’π‘›)+(π‘ƒβ„Žπ‘’π‘›βˆ’π‘’π‘›)=πœƒπ‘›+πœŒπ‘› into it, we obtain the following error formulation relating to πœƒπ‘› and πœŒπ‘›: ξ‚€πœ•π‘‘πœƒπ‘›ξ‚+ξ‚€π‘Žξ€·,π‘£Μƒπ‘’π‘›β„Žξ€Έπ‘Žπ·π›½π‘₯πœƒπ‘›ξ‚βˆ’ξ‚€π‘ξ€·,π·π‘£Μƒπ‘’π‘›β„Žξ€Έπ·πœƒπ‘›ξ‚=ξ‚€π‘Žξ€·,π‘£Μƒπ‘’π‘›β„Žξ€Έπ‘Žπ·π›½π‘₯πΌβ„Žπ‘’π‘›ξ‚+𝑏,π·π‘£Μƒπ‘’π‘›β„Žξ€Έπ‘Žπ·π›½π‘₯πΌβ„Žπ‘’π‘›ξ‚+ξ€·πœ•,π‘£π‘‘π‘’π‘›βˆ’1/2ξ€Έβˆ’ξ‚€,π‘£πœ•π‘‘πΌβ„Žπ‘’π‘›ξ‚+ξ‚€π‘Žξ€·π‘’,π‘£π‘›βˆ’1/2ξ€Έπ‘Žπ·π›½π‘₯π‘’π‘›βˆ’1/2ξ‚βˆ’ξ€·π‘ξ€·π‘’,π·π‘£π‘›βˆ’1/2ξ€Έπ·π‘’π‘›βˆ’1/2ξ€Έ+𝑓,π‘£Μƒπ‘’π‘›β„Žξ€Έξ€Έβˆ’ξ€·π‘“ξ€·π‘’,π‘£π‘›βˆ’1/2ξ€Έξ€Έξ‚€π‘Žξ€·,𝑣=βˆ’Μƒπ‘’π‘›β„Žξ€Έπ‘Žπ·π›½π‘₯πœŒπ‘›ξ‚+π‘Žξ€·π‘’,π·π‘£ξ‚†ξ‚€π‘›βˆ’1/2ξ€Έπ‘Žπ·π›½π‘₯π‘’π‘›βˆ’1/2ξ€·βˆ’π‘ŽΜƒπ‘’π‘›β„Žξ€Έπ‘Žπ·π›½π‘₯𝑒𝑛+𝑏,π·π‘£ξ‚ξ‚‡Μƒπ‘’π‘›β„Žξ€Έπ·πœŒπ‘›ξ€Έ+𝑏,π‘£ξ€½ξ€·Μƒπ‘’π‘›β„Žξ€Έπ·π‘’π‘›βˆ’ξ€·π‘ξ€·π‘’π‘›βˆ’1/2ξ€Έπ·π‘’π‘›βˆ’1/2+𝑓,π·π‘£ξ€Έξ€Ύξ€½ξ€·Μƒπ‘’π‘›β„Žξ€Έξ€·π‘’βˆ’π‘“π‘›βˆ’1/2ξ€Έ+πœ•,π‘£ξ€Έξ€Ύξ‚†ξ‚€π‘‘π‘’π‘›βˆ’1/2βˆ’πœ•π‘‘πΌβ„Žπ‘’π‘›,𝑣=𝑅1(𝑣)+𝑅2(𝑣)+𝑅3(𝑣)+𝑅4(𝑣)+𝑅5(𝑣)+𝑅6(𝑣).(3.35) Setting 𝑣=πœƒπ‘›, we obtain ξ‚€πœ•π‘‘πœƒπ‘›,πœƒπ‘›ξ‚+ξ‚€π‘Ž(̃𝑒𝑛)π‘Žπ·π›½π‘₯πœƒπ‘›,π·πœƒπ‘›ξ‚βˆ’ξ‚€π‘(̃𝑒𝑛)π·πœƒπ‘›,πœƒπ‘›ξ‚=𝑅1ξ‚€πœƒπ‘›ξ‚+𝑅2ξ‚€πœƒπ‘›ξ‚+𝑅3ξ‚€πœƒπ‘›ξ‚+𝑅4ξ‚€πœƒπ‘›ξ‚+𝑅5ξ‚€πœƒπ‘›ξ‚+𝑅6ξ‚€πœƒπ‘›ξ‚.(3.36) Note that ξ‚€πœ•π‘‘πœƒπ‘›,πœƒπ‘›ξ‚=ξ‚΅πœƒπ‘›βˆ’πœƒπ‘›βˆ’1𝜏,πœƒπ‘›+πœƒπ‘›βˆ’12ξ‚Ά=1ξ‚€2πœβ€–πœƒπ‘›β€–2βˆ’β€–β€–πœƒπ‘›βˆ’1β€–β€–2.(3.37) According to (3.2) and Lemma 3.2, we have ξ‚€π‘Ž(̃𝑒𝑛)π‘Žπ·π›½π‘₯πœƒπ‘›,π·πœƒπ‘›ξ‚|||β‰₯π‘šπœƒπ‘›|||2𝛼||πœƒβ‰₯𝑐𝑛||2𝛼+||πœƒπ‘›βˆ’1||2𝛼.(3.38) From Lemma 3.3, the following inequality can be derived: 𝑏(̃𝑒𝑛)πœƒπ‘›,π·πœƒπ‘›ξ‚β‰€π‘0πœ€βˆ’π‘12‖𝐷𝑏(̃𝑒𝑛)‖𝑐2β€–β€–πœƒπ‘›β€–β€–2+πœ€3β€–β€–πœƒπ‘›β€–β€–2𝛼=𝑐0πœ€βˆ’π‘12‖𝐷𝑏(̃𝑒𝑛)‖𝑐2β€–β€–β€–πœƒπ‘›+πœƒπ‘›βˆ’12β€–β€–β€–2+πœ€3β€–β€–β€–πœƒπ‘›+πœƒπ‘›βˆ’12β€–β€–β€–2𝛼≀𝑐3πœ€βˆ’π‘12‖𝐷𝑏(̃𝑒𝑛)‖𝑐2ξ‚€β€–πœƒπ‘›β€–2+β€–β€–πœƒπ‘›βˆ’1β€–β€–2+𝑐4πœ€3ξ‚€β€–πœƒπ‘›β€–2𝛼+β€–β€–πœƒπ‘›βˆ’1β€–β€–2𝛼.(3.39)
Substituting (3.37)–(3.39) into (3.36) then multiplying (3.36) by 2𝜏, summing from 𝑛=1 to 𝑁, we have β€–πœƒπ‘›β€–2βˆ’β€–β€–πœƒ2β€–β€–2+πœπ‘ξ“π‘›=1ξ€·2π‘šπ‘βˆ’2𝑐4πœ€3ξ€Έξ‚€β€–πœƒπ‘›β€–2𝛼+β€–β€–πœƒπ‘›βˆ’1β€–β€–2𝛼≀2πœπ‘ξ“π‘›=1𝑐3πœ€βˆ’π‘12β€–β€–ξ€·π·π‘Μƒπ‘’π‘›β„Žξ€Έβ€–β€–π‘2ξ‚€β€–πœƒπ‘›β€–2+β€–β€–πœƒπ‘›βˆ’1β€–β€–2+2πœπ‘ξ“π‘›=3𝑅1ξ‚€πœƒπ‘›ξ‚+𝑅2ξ‚€πœƒπ‘›ξ‚+𝑅3ξ‚€πœƒπ‘›ξ‚+𝑅4ξ‚€πœƒπ‘›ξ‚+𝑅5ξ‚€πœƒπ‘›ξ‚+𝑅6ξ‚€πœƒπ‘›.(3.40) We now estimate 𝑅1 to 𝑅6 in the right hand of (3.40), 𝑅1ξ‚€πœƒπ‘›ξ‚=ξ‚€π‘Žπ·π‘₯1βˆ’π›Όξ‚€π‘Žξ€·Μƒπ‘’π‘›β„Žξ€Έπ‘Žπ·π›½π‘₯πœŒπ‘›ξ‚,π‘Žπ·π›Όπ‘₯πœƒπ‘›ξ‚ξ‚€β‰€π‘€π‘Žπ·π›Όπ‘₯πœŒπ‘›,π‘Žπ·π›Όπ‘₯πœƒπ‘›ξ‚β€–β€–β‰€π‘€π‘Žπ·π›Όπ‘₯πœŒπ‘›β€–β€–β€–β€–π‘Žπ·π›Όπ‘₯πœƒπ‘›β€–β€–β‰€πœ€4β€–β€–πœƒπ‘›β€–β€–2𝛼+𝑐254πœ€4β€–β€–πœŒπ‘›β€–β€–2𝛼=πœ€42β€–β€–πœŒπ‘›+πœŒπ‘›βˆ’1β€–β€–2𝛼+𝑐2516πœ€4β€–β€–πœƒπ‘›+πœƒπ‘›βˆ’1β€–β€–2π›Όβ‰€πœ€4𝑐6ξ‚€β€–πœƒπ‘›β€–2𝛼+β€–β€–πœƒπ‘›βˆ’1β€–β€–2𝛼+𝑐7πœ€4ξ‚€β€–πœŒπ‘›β€–2𝛼+β€–β€–πœŒπ‘›βˆ’1β€–β€–2𝛼.(3.41)
Secondly, we deduce the estimation of 𝑅2, 𝑅2ξ‚€πœƒπ‘›ξ‚=ξ‚€ξ€·βˆ’π‘ŽΜƒπ‘’π‘›β„Žξ€Έπ‘Žπ·π›½π‘₯𝑒𝑛,π·πœƒπ‘›ξ‚+ξ‚€π‘Žξ€·π‘’π‘›βˆ’1/2ξ€Έπ‘Žπ·π›½π‘₯π‘’π‘›βˆ’1/2,π·πœƒπ‘›ξ‚=ξ‚€ξ€·π‘Žξ€·π‘’π‘›βˆ’1/2ξ€Έξ€·βˆ’π‘ŽΜƒπ‘’π‘›β„Žξ€Έξ€Έπ‘Žπ·π›½π‘₯𝑒𝑛,π·πœƒπ‘›ξ‚+ξ‚€π‘Žξ€·π‘’π‘›βˆ’1/2ξ€Έξ‚€π‘Žπ·π›½π‘₯π‘’π‘›βˆ’1/2βˆ’π‘Žπ·π›½π‘₯𝑒𝑛,π·πœƒπ‘›ξ‚=𝑅21+𝑅22,(3.42) where 𝑅21=ξ‚€ξ€Ίπ‘Žξ€·π‘’π‘›βˆ’1/2ξ€Έβˆ’π‘Ž(̃𝑒𝑛)ξ€»π‘Žπ·π›½π‘₯̃𝑒𝑛,π·πœƒπ‘›ξ‚β‰€π‘84πœ€5β€–β€–ξ€Ίπ‘Žξ€·π‘’π‘›βˆ’1/2ξ€Έβˆ’π‘Ž(̃𝑒𝑛)ξ€»π‘Žπ·π›½π‘₯̃𝑒𝑛‖‖21βˆ’π›Ό+πœ€5β€–β€–π·πœƒπ‘›β€–β€–2π›Όβˆ’1≀𝑐9β€–β€–π‘Žξ€·π‘’π‘›βˆ’1/2ξ€Έβˆ’π‘Ž(̃𝑒𝑛)β€–β€–β€–β€–π‘Žπ·π›½π‘₯̃𝑒𝑛‖‖21βˆ’π›Ό+πœ€5β€–β€–πœƒπ‘›β€–β€–2𝛼≀𝑐9πΏβ€–β€–π‘’π‘›βˆ’1/2βˆ’Μƒπ‘’π‘›β€–β€–+πœ€5β€–β€–πœƒπ‘›β€–β€–2𝛼,𝑅22=ξ‚€π‘Žξ€·π‘’π‘›βˆ’1/2ξ€Έξ‚€π‘Žπ·π›½π‘₯π‘’π‘›βˆ’1/2βˆ’π‘Žπ·π›½π‘₯𝑒𝑛,π·πœƒπ‘›ξ‚β‰€π‘104πœ€6β€–β€–π‘Žξ€·π‘’π‘›βˆ’1/2ξ€Έξ‚ƒπ‘Žπ·π›½π‘₯π‘’π‘›βˆ’1/2βˆ’π‘Žπ·π›½π‘₯𝑒𝑛‖‖21βˆ’π›Ό+πœ€6β€–β€–π·πœƒπ‘›β€–β€–2βˆ’1+𝛼≀𝑐10β€–β€–π‘Žξ€·π‘’π‘›βˆ’1/2ξ€Έβ€–β€–2β€–β€–π‘Žπ·π›½π‘₯π‘’π‘›βˆ’1/2βˆ’π‘Žπ·π›½π‘₯𝑒𝑛‖‖21βˆ’π›Ό+πœ€6β€–β€–πœƒπ‘›β€–β€–2𝛼≀𝑐10𝑀2β€–β€–π‘’π‘›βˆ’1/2βˆ’π‘’π‘›β€–β€–2𝛼+πœ€6π‘ξ‚€β€–πœƒπ‘›β€–2𝛼+β€–β€–πœƒπ‘›βˆ’1β€–β€–2𝛼.(3.43) The estimations of β€–Μƒπ‘’π‘›βˆ’π‘’π‘›βˆ’1/2β€– and β€–π‘’π‘›βˆ’π‘’π‘›βˆ’1/2‖𝛼 can be derived as follows: β€–β€–Μƒπ‘’π‘›βˆ’π‘’π‘›βˆ’1/2β€–β€–=β€–β€–β€–β€–32ξƒ¬π‘’π‘›βˆ’1/2βˆ’πœ2π‘’π‘‘π‘›βˆ’1/2+π‘’π‘›βˆ’1/2π‘‘π‘‘ξ‚€πœ2!22ξ€·πœ+𝑂3ξ€Έξƒ­βˆ’12ξƒ¬π‘’π‘›βˆ’1/2βˆ’3𝜏2π‘’π‘‘π‘›βˆ’1/2+π‘’π‘›βˆ’1/2𝑑𝑑2!3𝜏22ξ€·πœ+𝑂3ξ€Έξƒ­βˆ’π‘’π‘›βˆ’1/2‖‖‖‖≀𝑐11𝜏2β€–β€–π‘’π‘‘π‘‘ξ€·π‘‘π‘›βˆ’1/2‖‖≀𝑐11𝜏2ξ€œπ‘‘π‘›π‘‘π‘›βˆ’1‖‖𝑒𝑑𝑑‖‖‖‖(β‹…,𝑠)𝑑𝑠,π‘’π‘›βˆ’π‘’π‘›βˆ’1/2‖‖𝛼=β€–β€–β€–πœβˆ’1ξ‚»ξ€œπ‘‘π‘›π‘‘π‘›βˆ’1/2ξ€·π‘ βˆ’π‘‘π‘›ξ€Έ2π‘’π‘‘π‘‘ξ€œ(𝑠)𝑑𝑠+π‘‘π‘›βˆ’1/2π‘‘π‘›βˆ’1ξ€·π‘ βˆ’π‘‘π‘›βˆ’1ξ€Έ2𝑒𝑑𝑑‖‖‖(𝑠)𝑑𝑠𝛼≀𝑐12πœβ€–β€–β€–ξ€œπ‘‘π‘›π‘‘π‘›βˆ’1𝑒𝑑𝑑‖‖‖(𝑠)𝑑𝑠𝛼≀𝑐12πœξ€œπ‘‘π‘›π‘‘π‘›βˆ’1‖‖𝑒𝑑𝑑‖‖(𝑠)𝛼𝑑𝑠≀𝑐12πœβ„Žπ‘˜+1βˆ’π›Όξ€œπ‘‘π‘›π‘‘π‘›βˆ’1‖‖𝑒𝑑𝑑‖‖(𝑠)π‘˜+1𝑑𝑠.(3.44) Thirdly, it is turn to consider 𝑅3, 𝑅3ξ‚€πœƒπ‘›ξ‚=ξ‚€π‘ξ€·Μƒπ‘’π‘›β„Žξ€Έπ·πœŒπ‘›,πœƒπ‘›ξ‚β‰€β€–β€–π‘ξ€·Μƒπ‘’π‘›β„Žξ€Έπ·πœŒπ‘›β€–β€–βˆ’π›Όβ€–β€–πœƒπ‘›β€–β€–π›Όβ‰€π‘134πœ€7β€–β€–π‘ξ€·Μƒπ‘’π‘›β„Žξ€Έβ€–β€–2β€–β€–πœŒπ‘›β€–β€–21βˆ’π›Ό+πœ€7β€–β€–πœƒπ‘›β€–β€–2𝛼≀𝑐144πœ€7ξ‚€β€–πœŒπ‘›β€–21βˆ’π›Ό+β€–β€–πœŒπ‘›βˆ’1β€–β€–21βˆ’π›Όξ‚+πœ€7𝑐15ξ‚€β€–πœƒπ‘›β€–2𝛼+β€–β€–πœƒπ‘›βˆ’1β€–β€–2𝛼.(3.45) Next, 𝑅4ξ‚€πœƒπ‘›ξ‚=ξ‚€π‘ξ€·Μƒπ‘’π‘›β„Žξ€Έπ·π‘’π‘›,πœƒπ‘›ξ‚βˆ’ξ‚€π‘ξ€·π‘’π‘›βˆ’1/2ξ€Έπ·π‘’π‘›βˆ’1/2,πœƒπ‘›ξ‚=ξ‚€ξ€·π‘ξ€·Μƒπ‘’π‘›β„Žξ€Έπ·π‘’π‘›ξ€·π‘’βˆ’π‘π‘›βˆ’1/2𝐷𝑒𝑛,πœƒπ‘›ξ‚+ξ‚€π‘ξ€·π‘’π‘›βˆ’1/2π·ξ€Έξ€·π‘’π‘›βˆ’π·π‘’π‘›βˆ’1/2ξ€Έ,πœƒπ‘›ξ‚=𝑅41+𝑅42,(3.46) where 𝑅41≀𝑐164πœ€8β€–β€–ξ€½π‘ξ€·Μƒπ‘’π‘›β„Žξ€Έξ€·π‘’βˆ’π‘π‘›βˆ’1/2𝐷𝑒𝑛‖‖21βˆ’π›Ό+πœ€8β€–β€–πœƒπ‘›β€–β€–2𝛼≀𝑐16𝐿4πœ€8β€–β€–Μƒπ‘’π‘›β„Žβˆ’π‘’π‘›βˆ’1/2β€–β€–2‖‖𝐷𝑒𝑛‖‖21βˆ’π›Ό+πœ€8β€–β€–πœƒπ‘›β€–β€–2𝛼=𝑐16𝐿4πœ€8β€–β€–Μƒπ‘’π‘›β„Žβˆ’Μƒπ‘’π‘›+Μƒπ‘’π‘›βˆ’π‘’π‘›βˆ’1/2β€–β€–2‖‖𝐷𝑒𝑛‖‖21βˆ’π›Ό+πœ€8β€–β€–πœƒπ‘›β€–β€–2𝛼≀𝑐17β€–β€–Μƒπ‘’π‘›β„Žβˆ’Μƒπ‘’π‘›β€–β€–2+𝑐17β€–β€–Μƒπ‘’π‘›βˆ’π‘’π‘›βˆ’1/2β€–β€–2+πœ€8β€–β€–πœƒπ‘›β€–β€–2𝛼≀𝑐17β€–β€–Μƒπœƒπ‘›+ΜƒπœŒπ‘›β€–β€–2+𝑐17β€–β€–Μƒπ‘’π‘›βˆ’π‘’π‘›βˆ’1/2β€–β€–2+πœ€8β€–β€–πœƒπ‘›β€–β€–2𝛼≀𝑐18ξ‚€β€–β€–Μƒπœƒπ‘›β€–β€–2+β€–ΜƒπœŒπ‘›β€–2+𝑐17β€–β€–Μƒπ‘’π‘›βˆ’π‘’π‘›βˆ’1/2β€–β€–2+πœ€8β€–β€–πœƒπ‘›β€–β€–2𝛼.(3.47) Rewriting 𝑅42 by the aid of (3.20), we have 𝑅42≀𝑐194πœ€9β€–β€–π‘’π‘›βˆ’π‘’π‘›βˆ’1/2β€–β€–2+πœ€9β€–β€–πœƒπ‘›β€–β€–2𝛼.(3.48) The estimation of 𝑅5 is deduced as follows: 𝑅5ξ‚€πœƒπ‘›ξ‚β‰€β€–β€–π‘“ξ€·Μƒπ‘’π‘›β„Žξ€Έξ€·π‘’βˆ’π‘“π‘›βˆ’1/2ξ€Έβ€–β€–β€–β€–πœƒπ‘›β€–β€–β€–β€–β‰€πΏΜƒπ‘’π‘›β„Žβˆ’π‘’π‘›βˆ’1/2β€–β€–β€–β€–πœƒπ‘›β€–β€–β‰€πΏπ‘204πœ€10β€–β€–Μƒπ‘’π‘›β„Žβˆ’π‘’π‘›βˆ’1/2β€–β€–2+πœ€10β€–β€–πœƒπ‘›β€–β€–2≀𝐿𝑐21ξ‚€β€–β€–Μƒπœƒπ‘›+ΜƒπœŒπ‘›β€–β€–2+β€–β€–Μƒπ‘’π‘›βˆ’π‘’π‘›βˆ’1/2β€–β€–2+πœ€10β€–β€–πœƒπ‘›β€–β€–2≀𝑐22ξ‚€β€–β€–Μƒπœƒπ‘›β€–β€–2+β€–ΜƒπœŒπ‘›β€–2+𝐿𝑐21β€–β€–Μƒπ‘’π‘›βˆ’π‘’π‘›βˆ’1/2β€–β€–2+πœ€10β€–β€–πœƒπ‘›β€–β€–2.(3.49) Last, we estimate 𝑅6, 𝑅6ξ‚€πœƒπ‘›ξ‚=ξ‚€πœ•π‘‘π‘’π‘›βˆ’1/2,πœƒπ‘›ξ‚βˆ’ξ‚€πœ•π‘‘π‘ƒβ„Žπ‘’π‘›,πœƒπ‘›ξ‚=ξ‚€πœ•π‘‘π‘’π‘›βˆ’1/2βˆ’πœ•π‘’π‘›,πœƒπ‘›ξ‚+ξ‚€πœ•π‘‘π‘’π‘›βˆ’πœ•π‘‘π‘ƒβ„Žπ‘’π‘›,πœƒπ‘›ξ‚=ξ‚€πœ•π‘‘π‘’π‘›βˆ’1/2βˆ’πœ•π‘‘π‘’π‘›,πœƒπ‘›ξ‚+ξ‚€πœ•π‘‘πœŒπ‘›,πœƒπ‘›ξ‚β‰€β€–β€–πœ•π‘‘π‘’π‘›βˆ’1/2βˆ’πœ•π‘‘π‘’π‘›β€–β€–β€–πœƒπ‘›β€–β€–β€–+πœ•π‘‘πœŒπ‘›β€–β€–β€–πœƒπ‘›β€–,(3.50) where β€–β€–πœ•π‘‘π‘’π‘›βˆ’1/2βˆ’πœ•π‘’π‘›β€–β€–=(2𝜏)βˆ’1𝑐23β€–β€–β€–ξ€œπ‘‘π‘›π‘‘π‘›βˆ’1/2ξ€·π‘ βˆ’π‘‘π‘›ξ€Έ2𝑒𝑑𝑑𝑑(ξ€œπ‘ )𝑑𝑠+π‘‘π‘›βˆ’1/2π‘‘π‘›βˆ’1ξ€·π‘ βˆ’π‘‘π‘›βˆ’1ξ€Έ2𝑒𝑑𝑑𝑑(‖‖‖𝑠)𝑑𝑠≀𝑐23πœβ€–β€–β€–ξ€œπ‘‘π‘›π‘‘π‘›βˆ’1𝑒𝑑𝑑𝑑(‖‖‖𝑠)𝑑𝑠≀𝑐23πœξ€œπ‘‘π‘›π‘‘π‘›βˆ’1‖‖𝑒𝑑𝑑𝑑‖‖‖‖(𝑠)𝑑𝑠,πœ•π‘‘πœŒπ‘›β€–β€–=β€–β€–β€–πœŒπ‘›βˆ’πœŒπ‘›βˆ’1πœβ€–β€–β€–β‰€πœβˆ’1β€–β€–β€–ξ€œπ‘‘π‘›π‘‘π‘›βˆ’1πœŒπ‘›π‘‘β€–β€–β€–(𝑠)π‘‘π‘ β‰€πœβˆ’1ξ€œπ‘‘π‘›π‘‘π‘›βˆ’1‖‖𝑒𝑑‖‖(𝑠)π‘‘π‘ β‰€πœβˆ’1ξ€œπ‘‘π‘›π‘‘π‘›βˆ’1ξ€œ1π‘‘π‘ π‘‘π‘›π‘‘π‘›βˆ’1‖‖𝑒𝑑‖‖=ξ€œ(𝑠)π‘‘π‘ π‘‘π‘›π‘‘π‘›βˆ’1‖‖𝑒𝑑‖‖(𝑠)β‰€β„Žπ‘˜+1ξ€œπ‘‘π‘›π‘‘π‘›βˆ’1‖‖𝑒𝑑‖‖(𝑠)π‘˜+1𝑑𝑠.(3.51)
The β€–πœƒ2β€– should be estimated with (3.14). Let 𝑛=1 then subtracting (3.34) from the two equations of (3.14), respectively, one gets ξ‚€πœ•π‘‘πœƒ1,0+ξ‚€π‘Žξ€·π‘’,𝑣0β„Žξ€Έπ‘Žπ·π›½π‘₯πœƒ1,0𝑒,π·π‘£βˆ’π‘0β„Žξ€Έπ·πœƒ1,0ξ‚ξ‚€π‘Žξ€·π‘’,𝑣=βˆ’0β„Žξ€Έπ‘Žπ·π›½π‘₯𝜌1,0ξ‚βˆ’π‘Žξ€·π‘’,𝐷𝑣1/2ξ€Έπ‘Žπ·π›½π‘₯𝑒1/2ξ€·π‘’βˆ’π‘Ž0β„Žξ€Έπ‘Žπ·π›½π‘₯𝑒1,0+𝑏𝑒,𝐷𝑣0β„Žξ€Έπ·πœŒ1,0+𝑏𝑒,𝑣0β„Žξ€Έπ·π‘’1,0ξ€·π‘’βˆ’π‘1/2𝐷𝑒1/2+𝑓𝑒,𝐷𝑣0β„Žξ€Έξ€·π‘’βˆ’π‘“1/2ξ€Έ+πœ•,𝑣𝑑𝑒1/2βˆ’πœ•π‘‘πΌβ„Žπ‘’1,0,𝑣=𝑅1(𝑣)+𝑅2(𝑣)+𝑅3(𝑣)+𝑅4(𝑣)+𝑅5(𝑣)+𝑅6ξ‚€(𝑣),πœ•π‘‘πœƒ1+ξ‚€π‘Žξ€·π‘’,𝑣0β„Žξ€Έπ‘Žπ·π›½π‘₯πœƒ1𝑒,π·π‘£βˆ’π‘0β„Žξ€Έπ·πœƒ1ξ‚ξƒ©π‘Žξƒ©π‘’,𝑣=βˆ’0β„Ž+π‘’β„Ž1,02ξƒͺπ‘Žπ·π›½π‘₯𝜌1ξƒͺβˆ’π‘Žξ€·π‘’,𝐷𝑣1/2ξ€Έπ‘Žπ·π›½π‘₯𝑒1/2ξƒ©π‘’βˆ’π‘Ž0β„Ž+π‘’β„Ž1,02ξƒͺπ‘Žπ·π›½π‘₯𝑒1+𝑏𝑒,𝐷𝑣ξƒͺξƒ°0β„Ž+π‘’β„Ž1,02ξƒͺ𝐷𝜌1ξƒͺ+𝑏𝑒,𝑣0β„Ž+π‘’β„Ž1,02ξƒͺ𝐷𝑒1βˆ’ξ€·π‘ξ€·π‘’1/2𝐷𝑒1/2ξ€Έξƒ°+𝑓𝑒,𝐷𝑣0β„Ž+π‘’β„Ž1,02ξƒͺξ€·π‘’βˆ’π‘“1/2ξ€Έ+πœ•,𝑣ξƒͺ𝑑𝑒1/2βˆ’πœ•π‘‘πΌβ„Žπ‘’1,𝑣=𝑅1(𝑣)+𝑅2(𝑣)+𝑅3(𝑣)+𝑅4(𝑣)+𝑅5(𝑣)+𝑅6(𝑣).(3.52)
Setting 𝑣=πœƒ1,0, and using the similar estimation (see (3.40)), one has β€–β€–πœƒ1,0β€–β€–2ξ‚»πœβ‰€π‘2ξ€œπ‘‘1𝑑0‖‖𝑒𝑑𝑑(‖‖𝑠)2𝛼𝑑𝑠+β„Ž2(π‘˜+1βˆ’π›Ό)‖𝑒‖2π‘˜+1+𝜏2ξ€œπ‘‘1𝑑0‖‖𝑒𝑑𝑑𝑑‖‖(𝑠)2𝑑𝑠+π‘β„Ž2(π‘˜+1)ξ€œπ‘‘1𝑑0‖‖𝑒𝑑(‖‖𝑠)2π‘˜+1ξ‚Ό.𝑑𝑠(3.53)
Letting 𝑣=πœƒ1, applying the above result of πœƒ1,0, and using the similar estimation (see (3.53)), we get β€–β€–πœƒ1β€–β€–2ξ‚»πœβ‰€π‘2ξ€œπ‘‘1𝑑0‖‖𝑒𝑑𝑑(‖‖𝑠)2𝛼𝑑𝑠+β„Ž2(π‘˜+1βˆ’π›Ό)‖𝑒‖2π‘˜+1+𝜏2ξ€œπ‘‘1𝑑0‖‖𝑒𝑑𝑑𝑑(‖‖𝑠)2𝑑𝑠+π‘β„Ž2(π‘˜+1)ξ€œπ‘‘1𝑑0‖‖𝑒𝑑(‖‖𝑠)2π‘˜+1ξ‚Ό.𝑑𝑠(3.54)
Using 𝑇=π‘πœ and Gronwall’s lemma, we get β€–β€–||πœƒ||β€–β€–20,𝛼=𝑁𝑛=0πœβ€–πœƒβ€–2𝛼.(3.55)
Hence, using the interpolation property and β€–β€–||π‘’βˆ’π‘’β„Ž||β€–β€–0,𝛼≀‖‖||πœƒ||β€–β€–0,𝛼+β€–β€–||𝜌||β€–β€–0,𝛼,(3.56) the estimate (3.32) holds.
Also using the interpolation property, Gronwall’s lemma, and the approximation properties, we get β€–β€–||π‘’βˆ’π‘’β„Ž||β€–β€–βˆž,0≀‖‖||πœƒ||β€–β€–βˆž,0+β€–β€–||𝜌||β€–β€–βˆž,0≀max0≀𝑛≀𝑁‖‖||πœƒπ‘›||β€–β€–2+β„Ž2π‘˜+1β€–|𝑒|β€–2∞,π‘˜+1,(3.57) which is just the estimate (3.33).

4. Numerical Examples

In this section, we present the numerical results which confirm the theoretical analysis in Section 3.

Let 𝐾 denote a uniform partition on [0,π‘Ž], and π‘†β„Ž the space of continuous piecewise linear functions on 𝐾, that is, π‘˜=1.. In order to implement the Galerkin finite element approximation, we adapt finite element discrete along the space axis, and finite difference scheme along the time axis. We associate shape function of space π‘‹β„Ž with the standard basis of hat functions on the uniform grid of size β„Ž=1/𝑛. We have the predicted rates of convergence if the condition Δ𝑑=π‘β„Ž of β€–β€–π‘’βˆ’π‘’β„Žβ€–β€–0,π›Όξ€·β„ŽβˆΌπ‘‚2βˆ’π›Όξ€Έ,β€–β€–π‘’βˆ’π‘’β„Žβ€–β€–βˆž,0ξ€·β„ŽβˆΌπ‘‚2βˆ’π›Όξ€Έ,(4.1) provided that the initial value πœ‘(π‘₯) is smooth enough.

Example 4.1. The following equation πœ•π‘’ξ€·π‘’πœ•π‘‘=𝐷20𝐷π‘₯0.5𝑒(π‘₯,𝑑)βˆ’2π‘₯(π‘₯βˆ’1)2π‘₯1.5βˆ’π‘₯Ξ“(2.5)0.5𝑒Γ(1.5)βˆ’2π‘‘π·π‘’βˆ’π‘’(π‘₯,𝑑)βˆ’π‘’2π‘’βˆ’π‘‘ξ‚΅2π‘₯0.5βˆ’π‘₯Ξ“(1.5)βˆ’0.5ξ‚ΆΞ“(0.5),0≀π‘₯≀1,0≀𝑑≀1,𝑒(π‘₯,0)=π‘₯(π‘₯βˆ’1),0≀π‘₯≀1,𝑒(0,𝑑)=𝑒(1,𝑑)=0,0≀𝑑≀1,(4.2) has a unique solution 𝑒(π‘₯,𝑑)=π‘’βˆ’π‘‘π‘₯(π‘₯βˆ’1).
If we select Δ𝑑=π‘β„Ž and note that the initial value 𝑒0 is smooth enough, then we have β€–β€–π‘’βˆ’π‘’β„Žβ€–β€–0,0.75ξ€·β„ŽβˆΌπ‘‚1.25ξ€Έ,β€–β€–π‘’βˆ’π‘’β„Žβ€–β€–βˆž,0ξ€·β„ŽβˆΌπ‘‚1.25ξ€Έ.(4.3)

Table 1 includes numerical calculations over a regular partition of [0,1]. We can observe the experimental rates of convergence agree with the theoretical rates for the numerical solution.


β„Ž β€– 𝑒 βˆ’ 𝑒 β„Ž β€– ∞ , 0 cvge. rate β€– 𝑒 βˆ’ 𝑒 β„Ž β€– 0 , 0 . 7 5 cvge. rate

1/52.2216Eβˆ’003 β€”1.0213Eβˆ’003 β€”
1/101.3551Eβˆ’003 0.71326.0779Eβˆ’004 0.74875
1/205.5865Eβˆ’004 1.27842.3188Eβˆ’004 1.3901
1/403.0515Eβˆ’004 0.87241.0545Eβˆ’004 1.1367
1/801.2423Eβˆ’004 1.29643.9883Eβˆ’005 1.4027
1/1605.1033Eβˆ’005 1.28352.1310Eβˆ’005 0.9042

Example 4.2. The function 𝑒(π‘₯,𝑑)=cos(𝑑)π‘₯2(2βˆ’π‘₯)2 solves the equation in the following form: πœ•π‘’=πœ•π‘‘0𝐷π‘₯1.7[][𝑒(π‘₯,𝑑)+𝑏(𝑒)π·π‘’βˆ’π‘’4(1βˆ’π‘₯)+tan𝑑+𝑓(π‘₯,𝑑),π‘₯∈(0,2),π‘‘βˆˆ0,1),𝑒(π‘₯,0)=π‘₯2(2βˆ’π‘₯)2𝑒,0≀π‘₯≀2,(0,𝑑)=0,𝑒(2,𝑑)=0,0≀𝑑≀1,(4.4) where βˆšπ‘(𝑒)=π‘’βˆš,cos𝑑𝑓(π‘₯,𝑑)=cos𝑑π‘₯cos(0.85πœ‹)242.3+(2βˆ’π‘₯)2.3ξ€Έβˆ’ξ€·π‘₯Ξ“(3.3)241.3+(2βˆ’π‘₯)1.3ξ€Έβˆ’8ξ€·π‘₯Ξ“(2.3)0.3+(2βˆ’π‘₯)0.3ξ€Έξƒ­.Ξ“(1.3)(4.5)
If we select Δ𝑑=π‘β„Ž, then β€–β€–π‘’βˆ’π‘’β„Žβ€–β€–0,0.85ξ€·β„ŽβˆΌπ‘‚1.15ξ€Έ,β€–β€–π‘’βˆ’π‘’β„Žβ€–β€–βˆž,0ξ€·β„ŽβˆΌπ‘‚1.15ξ€Έ.(4.6)

Table 2 shows the error results at different size of space grid. We can observe that the experimental rates of convergence still support the theoretical rates.


β„Ž β€– 𝑒 βˆ’ 𝑒 β„Ž β€– ∞ , 0 cvge. rate β€– 𝑒 βˆ’ 𝑒 β„Ž β€– 0 , 0 . 8 5 cvge. rate

1/5 1.3010Eβˆ’001 β€”3.2223Eβˆ’002 β€”
1/104.6402Eβˆ’002 1.48781.4133Eβˆ’002 1.1890
1/201.6843Eβˆ’002 1.46206.2946Eβˆ’003 1.1669
1/406.6019Eβˆ’003 1.68432.8571Eβˆ’003 1.1395
1/802.7979Eβˆ’003 1.23861.3137Eβˆ’003 1.1209
1/1601.2665Eβˆ’003 1.14346.0848Eβˆ’004 1.1103

Example 4.3. Consider the following space-fractional differential equation with the nonhomogeneous boundary conditions, πœ•π‘’=πœ•π‘‘0𝐷π‘₯1.73𝑒(π‘₯,𝑑)βˆ’π‘₯ξ€œπ‘₯0𝑒𝑑π‘₯βˆ’2π‘₯0.3π‘’βˆ’π‘‘Ξ“(1.3),0≀π‘₯≀1,0≀𝑑≀1,𝑒(π‘₯,0)=π‘₯2,0≀π‘₯≀1,𝑒(0,𝑑)=0,𝑒(1,𝑑)=π‘’βˆ’π‘‘,0≀𝑑≀1,(4.7) whose exact solution is 𝑒(π‘₯,𝑑)=π‘’βˆ’π‘‘π‘₯2.
We still choose Δ𝑑=π‘β„Ž, then get the convergence rates β€–β€–π‘’βˆ’π‘’β„Žβ€–β€–0,0.85ξ€·β„ŽβˆΌπ‘‚1.15ξ€Έ,β€–β€–π‘’βˆ’π‘’β„Žβ€–β€–βˆž,0ξ€·β„ŽβˆΌπ‘‚1.15ξ€Έ.(4.8)

The numerical results are presented in Table 3 which are in line with the theoretical analysis.


β„Ž β€– 𝑒 βˆ’ 𝑒 β„Ž β€– ∞ , 0 cvge. rate β€– 𝑒 βˆ’ 𝑒 β„Ž β€– 0 , 0 . 8 5 cvge. rate

1/58.3052Eβˆ’002 β€”2.8009Eβˆ’002 β€”
1/103.6038Eβˆ’002 1.20451.0086Eβˆ’002 1.4735
1/201.3839Eβˆ’002 1.38073.2327Eβˆ’003 1.6414
1/405.0631Eβˆ’003 1.45071.1789Eβˆ’003 1.4554
1/801.8920Eβˆ’003 1.42015.9555Eβˆ’004 0.9851
1/1609.9899Eβˆ’004 0.92143.0034Eβˆ’004 0.9876

5. Conclusion

In this paper, we propose a fully discrete Galerkin finite element method to solve a type of fractional advection-diffusion equation numerically. In the temporal direction we use the modified Crank-Nicolson method, and in the spatial direction we use the finite element method. The error analysis is derived on the basis of fractional derivative space. The numerical results agree with the theoretical error estimates, demonstrating that our algorithm is feasible.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China under grant no. 10872119, the Key Disciplines of Shanghai Municipality under grant no. S30104, the Key Program of Shanghai Municipal Education Commission under grant no. 12ZZ084, and the Natural Science Foundation of Anhui province KJ2010B442.

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Copyright Β© 2011 Yunying Zheng and Zhengang Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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