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Abstract and Applied Analysis
Volumeย 2012ย (2012), Article IDย 596184, 25 pages
http://dx.doi.org/10.1155/2012/596184
Research Article

Finite Element Solutions for the Space Fractional Diffusion Equation with a Nonlinear Source Term

Department of Mathematics Education, Seoul National University, Seoul 151-748, Republic of Korea

Received 26 April 2012; Revised 6 July 2012; Accepted 17 July 2012

Academic Editor: Bashirย Ahmad

Copyright ยฉ 2012 Y. J. Choi and S. K. Chung. 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.

Abstract

We consider finite element Galerkin solutions for the space fractional diffusion equation with a nonlinear source term. Existence, stability, and order of convergence of approximate solutions for the backward Euler fully discrete scheme have been discussed as well as for the semidiscrete scheme. The analytical convergent orders are obtained as ๐‘‚(๐‘˜+โ„Žฬƒ๐›พ), where ฬƒ๐›พ is a constant depending on the order of fractional derivative. Numerical computations are presented, which confirm the theoretical results when the equation has a linear source term. When the equation has a nonlinear source term, numerical results show that the diffusivity depends on the order of fractional derivative as we expect.

1. Introduction

Fractional calculus is an old mathematical topic but it has not been attracted enough for almost three hundred years. However, it has been recently proven that fractional calculus is a significant tool in the modeling of many phenomena in various fields such as engineering, physics, porous media, economics, and biological sciences. One can see related references in [1โ€“7].

In the classical diffusion model, it is assumed that particles are distributed in a normal bell-shaped pattern based on the Brownian motion. In general, the nature of diffusion is characterized by the mean squared displacement ๎ซ(โ–ต๐‘Ÿ)2๎ฌ=2๐‘‘๐œ…๐œ‡๐‘ก๐œ‡,(1.1) where ๐‘‘ is the spatial dimension and ๐œ…๐œ‡ is the diffusion constant. The classical normal diffusion case arises when the exponent ๐œ‡=1. When ๐œ‡โ‰ 1, anomalous diffusions arise. The anomalous diffusion is classified as the process is subdiffusive (diffusive slowly) when ๐œ‡<1 or superdiffusive (diffusive fast) when ๐œ‡>1.

As mentioned before, in many real problems, it is more adequate to use anomalous diffusion described by fractional derivatives than the classical normal diffusion [4, 5, 8โ€“12]. One typical model for anomalous diffusion is the fractional superdiffusion equation arising in chaotic and turbulent processes, where the usual second derivative in space is replaced by a fractional derivative of order 1<๐œ‡<2.

In this paper we discuss Galerkin approximate solutions for the space fractional diffusion equation with a nonlinear source term. The equation is described as ๐œ•๐‘ข(๐‘ฅ,๐‘ก)๐œ•๐‘ก=๐œ…๐œ‡โˆ‡๐œ‡๐‘ข(๐‘ฅ,๐‘ก)+๐‘“(๐‘ฅ,๐‘ก,๐‘ข)(1.2) with an initial condition ๐‘ข(๐‘ฅ,0)=๐‘ข0,๐‘ฅโˆˆฮฉโŠ‚๐‘(1.3) and boundary conditions ๐‘ข(๐‘ฅ,๐‘ก)=0,๐‘ฅโˆˆ๐œ•ฮฉ,0โ‰ค๐‘กโ‰ค๐‘‡,(1.4) where ๐œ…๐œ‡ denotes the anomalous diffusion coefficient and ๐œ•ฮฉ is the boundary of the domain ฮฉ. And the differential operator โˆ‡๐œ‡ is โˆ‡๐œ‡=12๐‘Ž๐ท๐œ‡๐‘ฅ+12๐‘ฅ๐ท๐œ‡๐‘,(1.5) where ๐‘Ž๐ท๐œ‡๐‘ฅ and ๐‘ฅ๐ท๐œ‡๐‘ are called the left and the right Riemann-Liouville space fractional derivatives of order ๐œ‡, respectively, defined by ๐ƒ๐œ‡๐‘ขโˆถ=๐‘Ž๐ท๐œ‡๐‘ฅ๐‘ข(๐‘ฅ)=๐ท๐‘›๐‘Ž๐ท๐‘ฅ๐œ‡โˆ’๐‘›1๐‘ข(๐‘ฅ)=๐‘‘ฮ“(๐‘›โˆ’๐œ‡)๐‘›๐‘‘๐‘ฅ๐‘›๎€œ๐‘ฅ๐‘Ž(๐‘ฅโˆ’๐œ‰)๐‘›โˆ’๐œ‡โˆ’1๐ƒ๐‘ข(๐œ‰)๐‘‘๐œ‰,๐œ‡โˆ—๐‘ขโˆถ=๐‘ฅ๐ท๐œ‡๐‘๐‘ข(๐‘ฅ)=(โˆ’๐ท)๐‘›๐‘ฅ๐ท๐‘๐œ‡โˆ’๐‘›๐‘ข(๐‘ฅ)=(โˆ’1)๐‘›ฮ“๐‘‘(๐‘›โˆ’๐œ‡)๐‘›๐‘‘๐‘ฅ๐‘›๎€œ๐‘๐‘ฅ(๐œ‰โˆ’๐‘ฅ)๐‘›โˆ’๐œ‡โˆ’1๐‘ข(๐œ‰)๐‘‘๐œ‰.(1.6) Here ๐‘› is the smallest integer such that ๐‘›โˆ’1โ‰ค๐œ‡<๐‘›.

Throughout this paper, we will assume that the nonlinear source term ๐‘“(๐‘ฅ,๐‘ก,๐‘ข) is locally Lipschitz continuous with constants ๐ถ๐‘™ and ๐ถ๐‘“ such that โ€–๐‘“(๐‘ข)โˆ’๐‘“(๐‘ฃ)โ€–๐ฟ2(ฮฉ)โ‰ค๐ถ๐‘™โ€–๐‘ขโˆ’๐‘ฃโ€–๐ฟ2(ฮฉ),(1.7)โ€–๐‘“(๐‘ข)โ€–๐ฟ2(ฮฉ)โ‰ค๐ถ๐‘“โ€–๐‘ขโ€–๐ฟ2(ฮฉ)(1.8) for ๐‘ข,๐‘ฃโˆˆ{๐‘คโˆˆ๐ป0๐œ‡/2(ฮฉ)โˆฃโ€–๐‘คโ€–๐ฟ2(ฮฉ)โ‰ค๐‘™}.

Baeumer et al. [8, 13] have proved existence and uniqueness of a strong solution for (1.2) using the semigroup theory when ๐‘“(๐‘ฅ,๐‘ก,๐‘ข) is globally Lipschitz continuous. Furthermore, when ๐‘“(๐‘ฅ,๐‘ก,๐‘ข) is locally Lipschitz continuous, existence of a unique strong solution has also been shown by introducing the cut-off function.

Finite difference methods have been studied in [14โ€“16] for linear space fractional diffusion problems. They used the right-shifted Grรผwald-Letnikov approximate for the fractional derivative since the standard Grรผwald-Letnikov approximate gives the unconditional instability even for the implicit method. Using the right-shifted Grรผwald-Letnikov approximation, the method of lines has been applied in [12] for numerical approximate solutions.

For the space fractional diffusion problems with a nonlinear source term, Lynch et al. [17] used the so-called L2 and L2C methods in [6] and compared computational accuracy of them. Baeumer et al. [8] give existence of the solution and computational results using finite difference methods. Choi et al. [18] have shown existence and stability of numerical solutions of an implicit finite difference equation obtained by using the right-shifted Grรผwald-Letnikov approximation. For the time fractional diffusion equations, explicit and implicit finite difference methods have been used in [11, 19โ€“23].

Compared to finite difference methods on the fractional diffusion equation, finite element methods have been rarely discussed. Ervin and Roop [24] have considered finite element analysis for stationary linear advection dispersion equations, and Roop [25] has studied finite element analysis for nonstationary linear advection dispersion equations. The finite element numerical approximations have been discussed for the time and space fractional Fokker-Planck equation in Deng [9] and for the space general fractional diffusion equations with a nonlocal quadratic nonlinearity but a linear source term in Ervin et al. [26].

As far as we know, finite element methods have not been considered for the space fractional diffusion equation with nonlinear source terms. In this paper, we will discuss finite element solutions for the problem (1.2)โ€“(1.4) under the assumption of existence of a sufficiently regular solution ๐‘ข of the equation. Finite element numerical analysis of the semidiscrete and fully discrete methods for (1.2)โ€“(1.4) will be considered using the backward Euler method in time and Galerkin finite element method in space as well as the semidiscrete method. We will discuss existence, uniqueness, and stability of the numerical solutions for the problem (1.2)โ€“(1.4). Also, ๐ฟ2-error estimate will be considered for the problem (1.2)โ€“(1.4).

The outline of the paper is as follows. We introduce some properties of the space fractional derivatives in Section 2, which will be used in later discussion. In Section 3, the semidiscrete variational formulation for (1.2) based on Galerkin method is given. Existence, stability and ๐ฟ2-error estimate of the semidiscrete solution are analyzed. In Section 4, existence and unconditional stability of approximate solutions for the fully discrete backward Euler method are shown following the idea of the semidiscrete method. Further, ๐ฟ2-error estimates are obtained, whose convergence is of )๐‘‚(๐‘˜+โ„Žฬƒ๐›พ, where ฬƒ๐›พ=๐œ‡ if ๐œ‡โ‰ 3/2 and ฬƒ๐›พ=๐œ‡โˆ’๐œ–,0<๐œ–<1/2, if ๐œ‡=3/2. Finally, numerical examples are given in order to see the theoretical convergence order discussed in Section 5. We will see that numerical solutions of fractional diffusion equations diffuse more slowly than that of the classical diffusion problem and diffusivity depends on the order of fractional derivatives.

2. The Variational Form

In this section we will consider the variational form of problem (1.2)โ€“(1.4) and show existence and stability of the weak solution. We first recall some basic properties of Riemann-Liouville fractional calculus [9, 24].

For any given positive number ๐œ‡>0, define the seminorm |๐‘ข|๐ฝ๐œ‡๐ฟ(๐‘)=โ€–๐ƒ๐œ‡๐‘ขโ€–๐ฟ2(๐‘)(2.1) and the norm โ€–๐‘ขโ€–๐ฝ๐œ‡๐ฟ(๐‘)=๎‚€โ€–๐‘ขโ€–2๐ฟ2(๐‘)+|๐‘ข|2๐ฝ๐œ‡๐ฟ(๐‘)๎‚1/2,(2.2) where the left fractional derivative space ๐ฝ๐œ‡๐ฟ(๐‘) denotes the closure of ๐ถโˆž0(๐‘) with respect to the norm โ€–โ‹…โ€–๐ฝ๐œ‡๐ฟ(๐‘).

Similarly, we may define the right fractional derivative space ๐ฝ๐œ‡๐‘…(๐‘) as the closure of ๐ถโˆž0(๐‘) with respect to the norm โ€–โ‹…โ€–๐ฝ๐œ‡๐‘…(๐‘), where โ€–๐‘ขโ€–๐ฝ๐œ‡๐‘…(๐‘)=๎‚€โ€–๐‘ขโ€–2๐ฟ2(๐‘)+|๐‘ข|2๐ฝ๐œ‡๐‘…(๐‘)๎‚1/2(2.3) and the seminorm |๐‘ข|๐ฝ๐œ‡๐‘…(๐‘)=โ€–โ€–๐ƒ๐œ‡โˆ—๐‘ขโ€–โ€–๐ฟ2(๐‘).(2.4)

Furthermore, with the help of Fourier transform we define a seminorm |๐‘ข|๐ป๐œ‡(๐‘)=โ€–|๐œ”|๐œ‡ฬ‚๐‘ขโ€–๐ฟ2(๐‘)(2.5) and the norm โ€–๐‘ขโ€–๐ป๐œ‡(๐‘)=๎‚€โ€–๐‘ขโ€–2๐ฟ2(๐‘)+|๐‘ข|2๐ป๐œ‡(๐‘)๎‚1/2.(2.6) Here ๐ป๐œ‡(๐‘) denotes the closure of ๐ถโˆž0(๐‘) with respect to โ€–โ‹…โ€–๐ป๐œ‡(๐‘). It is known in [24] that the spaces ๐ฝ๐œ‡๐ฟ(๐‘),๐ฝ๐œ‡๐‘…(๐‘), and ๐ป๐œ‡(๐‘) are all equal with equivalent seminorms and norms. Analogously, when the domain ฮฉ is a bounded interval, the spaces ๐ฝ๐œ‡๐ฟ,0(ฮฉ),๐ฝ๐œ‡๐‘…,0(ฮฉ), and ๐ป๐œ‡0(ฮฉ) are equal with equivalent seminorms and norms [24, 27].

The following lemma on the Riemann-Liouville fractional integral operators will be used in our analysis, which can be proved by using the property of Fourier transform [24].

Lemma 2.1. For a given ๐œ‡>0 and a real valued function ๐‘ข๎€ท๐ƒ๐œ‡๐‘ข,๐ƒ๐œ‡โˆ—๐‘ข๎€ธ=cos(๐œ‹๐œ‡)โ€–๐ƒ๐œ‡๐‘ขโ€–2๐ฟ2(๐‘).(2.7)

Remark 2.2. It follows from (2.7) that we may use the following norm: โ€–๐‘ขโ€–2๐ป0๐œ‡/2(๐‘)=โ€–๐‘ขโ€–2๐ฟ2(๐‘)+๐œ…๐œ‡|||๎‚€๐œ‡cos๐œ‹โ‹…2๎‚||||๐‘ข|2๐ป0๐œ‡/2(๐‘)(2.8) instead of the norm โ€–๐‘ขโ€–๐ป๐œ‡(๐‘).

For the seminorm on ๐ป๐œ‡0(ฮฉ) with ฮฉ=(๐‘Ž,๐‘), the following fractional Poincarรฉ-Friedrichโ€™s inequality holds. For the proof, we refer to [9, 24].

Lemma 2.3. For ๐‘ขโˆˆ๐ป๐œ‡0(ฮฉ), there is a positive constant ๐ถ such that โ€–๐‘ขโ€–๐ฟ2(ฮฉ)โ‰ค๐ถ|๐‘ข|๐ป๐œ‡0(ฮฉ)(2.9) and for 0<๐‘ <๐œ‡,๐‘ โ‰ ๐‘›โˆ’1/2,๐‘›โˆ’1โ‰ค๐œ‡<๐‘›,๐‘›โˆˆโ„•, |๐‘ข|๐ป๐‘ 0(ฮฉ)โ‰ค๐ถ|๐‘ข|๐ป๐œ‡0(ฮฉ).(2.10)

Hereafter, a positive number ๐ถ will denote a generic constant. Also the semigroup property and the adjoint property hold for the Riemann-Liouville fractional integral operators [9, 24]: for all ๐œ‡,๐œˆ>0, if ๐‘ขโˆˆ๐ฟ๐‘(ฮฉ), ๐‘โ‰ฅ1, then ๐‘Ž๐ท๐‘ฅ๐‘Žโˆ’๐œ‡๐ท๐‘ฅโˆ’๐œˆ๐‘ข(๐‘ฅ)=๐‘Ž๐ท๐‘ฅโˆ’๐œ‡โˆ’๐œˆ๐‘ข(๐‘ฅ),โˆ€๐‘ฅโˆˆฮฉ,๐‘ฅ๐ท๐‘๐‘ฅโˆ’๐œ‡๐ท๐‘โˆ’๐œˆ๐‘ข(๐‘ฅ)=๐‘ฅ๐ท๐‘โˆ’๐œ‡โˆ’๐œˆ๐‘ข(๐‘ฅ),โˆ€๐‘ฅโˆˆฮฉ,(2.11) and specially ๎€ท๐‘Ž๐ท๐‘ฅโˆ’๐œ‡๎€ธ๐‘ข,๐‘ฃ๐ฟ2(ฮฉ)=๎€ท๐‘ข,๐‘ฅ๐ท๐‘โˆ’๐œ‡๐‘ฃ๎€ธ๐ฟ2(ฮฉ),โˆ€๐‘ข,๐‘ฃโˆˆ๐ฟ2(ฮฉ).(2.12)

In the rest of this section, we will consider a weak problem for (1.2)โ€“(1.4) with 1<๐œ‡<2: find a function ๐‘ขโˆˆ๐ป0๐œ‡/2(ฮฉ) such that ๎€ท๐‘ข๐‘ก๎€ธ=๎€ท๐œ…,๐‘ฃ๐œ‡โˆ‡๐œ‡๎€ธ๐‘ข,๐‘ฃ+(๐‘“(๐‘ข),๐‘ฃ),โˆ€๐‘ฃโˆˆ๐ป0๐œ‡/2(ฮฉ).(2.13)

Since there is a weak solution of (2.13) when ๐‘“ is locally Lipschitz continuous as in [8, 13], we here only discuss the stability of the weak solution, to show that we need the following lemma.

Lemma 2.4. For all ๐‘ฃโˆˆ๐ป0๐œ‡/2(ฮฉ), the following inequality holds: โˆ’๎€ท๐œ…๐œ‡โˆ‡๐œ‡๎€ธ๐‘ฃ,๐‘ฃโ‰ฅ๐œ…๐œ‡|||๎‚€๐œ‡cos๐œ‹โ‹…2๎‚||||๐‘ฃ|2๐ป0๐œ‡/2(ฮฉ).(2.14)

Proof. Following the ideas in [9, 26], we obtain the following inequality by using the properties (2.11)-(2.12) and Lemmas 2.1โ€“2.3: โˆ’๎€ท๐œ…๐œ‡โˆ‡๐œ‡๎€ธ๐œ…๐‘ฃ,๐‘ฃ=โˆ’๐œ‡2๎€ฝ๎€ท๐‘Ž๐ท๐œ‡๐‘ฅ๎€ธ+๎€ท๐‘ฃ,๐‘ฃ๐‘ฅ๐ท๐œ‡๐‘๐œ…๐‘ฃ,๐‘ฃ๎€ธ๎€พ=โˆ’๐œ‡2๎‚ป๎€œ๐‘๐‘Ž๎‚€๐ท2๐‘Ž๐ท๐‘ฅโˆ’(2โˆ’๐œ‡)๐‘ฃ๎‚๎€œ๐‘ฃ๐‘‘๐‘ฅ+๐‘๐‘Ž๎‚€(โˆ’๐ท)2๐‘ฅ๐ท๐‘โˆ’(2โˆ’๐œ‡)๐‘ฃ๎‚๎‚ผ=๐œ…๐‘ฃ๐‘‘๐‘ฅ๐œ‡2๎‚ป๎€œ๐‘๐‘Ž๎‚€๐ท๐‘Ž๐ท๐‘ฅโˆ’(2โˆ’๐œ‡)๐‘ฃ๎‚๎€œ๐ท๐‘ฃ๐‘‘๐‘ฅ+๐‘๐‘Ž๎‚€๐ท๐‘ฅ๐ท๐‘โˆ’(2โˆ’๐œ‡)๐‘ฃ๎‚๎‚ผ=๐œ…๐ท๐‘ฃ๐‘‘๐‘ฅ๐œ‡2๎‚ป๎€œ๐‘๐‘Ž๎‚€๐‘Ž๐ท๐‘ฅโˆ’(2โˆ’๐œ‡)๎‚๎€œ๐ท๐‘ฃ๐ท๐‘ฃ๐‘‘๐‘ฅ+๐‘๐‘Ž๎‚€๐‘ฅ๐ท๐‘โˆ’(2โˆ’๐œ‡)๎‚๎‚ผ=๐œ…๐ท๐‘ฃ๐ท๐‘ฃ๐‘‘๐‘ฅ๐œ‡2๎‚ป๎€œ๐‘๐‘Ž๎‚€๐‘Ž๐ท๐‘ฅ๐‘Žโˆ’(2โˆ’๐œ‡)/2๐ท๐‘ฅโˆ’(2โˆ’๐œ‡)/2๎‚+๎€œ๐ท๐‘ฃ๐ท๐‘ฃ๐‘‘๐‘ฅ๐‘๐‘Ž๎‚€๐‘ฅ๐ท๐‘๐‘ฅโˆ’((2โˆ’๐œ‡)/2)๐ท๐‘โˆ’(2โˆ’๐œ‡)/2๎‚๎‚ผ=๐œ…๐ท๐‘ฃ๐ท๐‘ฃ๐‘‘๐‘ฅ๐œ‡2๎‚ป๎€œ๐‘๐‘Ž๎‚€๐‘Ž๐ท๐‘ฅโˆ’(2โˆ’๐œ‡)/2๐ท๐‘ฃ๎‚๎‚€๐‘ฅ๐ท๐‘โˆ’(2โˆ’๐œ‡)/2๎‚+๎€œ๐ท๐‘ฃ๐‘‘๐‘ฅ๐‘๐‘Ž๎‚€๐‘ฅ๐ท๐‘โˆ’(2โˆ’๐œ‡)/2๐ท๐‘ฃ๎‚๎‚€๐‘Ž๐ท๐‘ฅโˆ’(2โˆ’๐œ‡)/2๎‚๎‚ผ๐ท๐‘ฃ๐‘‘๐‘ฅ=โˆ’๐œ…๐œ‡๎€ท๐ƒ๐œ‡/2๐‘ฃ,๐ƒ(๐œ‡/2)โˆ—๐‘ฃ๎€ธ=โˆ’๐œ…๐œ‡๎‚€๐œ‡cos๐œ‹โ‹…2๎‚โ€–โ€–๐ƒ๐œ‡/2๐‘ฃโ€–โ€–2๐ฟ2(ฮฉ)โ‰ฅ๐œ…๐œ‡|||๎‚€๐œ‡cos๐œ‹โ‹…2๎‚||||๐‘ฃ|2๐ป0๐œ‡/2(ฮฉ).(2.15) This completes the proof.

We consider the stability of a weak solution ๐‘ข for (2.13).

Theorem 2.5. Let ๐‘ข be a solution of (2.13). Then there is a constant ๐ถ such that โ€–๐‘ข(๐‘ก)โ€–๐ฟ2(ฮฉ)โ‰ค๐ถโ€–๐‘ข(0)โ€–๐ฟ2(ฮฉ).(2.16)

Proof. Taking ๐‘ฃ=๐‘ข(๐‘ก) in (2.13), we obtain ๎€ท๐‘ข๐‘ก๎€ธโˆ’๎€ท๐œ…,๐‘ข๐œ‡โˆ‡๐œ‡๎€ธ=๐‘ข,๐‘ข(๐‘“(๐‘ข),๐‘ข).(2.17) Since the second term on the left hand side is nonnegative from Lemma 2.4, we have 12๐‘‘๐‘‘๐‘กโ€–๐‘ขโ€–2๐ฟ2(ฮฉ)โ‰ค12๐‘‘๐‘‘๐‘กโ€–๐‘ขโ€–2๐ฟ2(ฮฉ)+๐œ…๐œ‡|||๎‚€๐œ‡cos๐œ‹โ‹…2๎‚||||๐‘ข|2๐ป0๐œ‡/2(ฮฉ)โ‰คโ€–๐‘“(๐‘ข)โ€–๐ฟ2(ฮฉ)โ€–๐‘ขโ€–๐ฟ2(ฮฉ)โ‰ค๐ถ๐‘“โ€–๐‘ขโ€–2๐ฟ2(ฮฉ).(2.18) Integrating both sides with respect to ๐‘ก, we obtain โ€–๐‘ข(๐‘ก)โ€–2๐ฟ2(ฮฉ)โ‰คโ€–๐‘ข(0)โ€–2๐ฟ2(ฮฉ)๎€œ+๐ถ๐‘ก0โ€–๐‘ข(๐‘ )โ€–2๐ฟ2(ฮฉ)๐‘‘๐‘ .(2.19) An application of Gronwall's inequality gives that there is a constant ๐ถ such that โ€–๐‘ข(๐‘ก)โ€–2๐ฟ2(ฮฉ)โ‰ค๐ถโ€–๐‘ข(0)โ€–2๐ฟ2(ฮฉ).(2.20) This completes the proof.

3. The Semidiscrete Variational Form

In this section, we will analyze the stability and error estimates of Galerkin finite element solutions for the semidiscrete variational formulation for (1.2).

Let ๐‘†โ„Ž be a partition of ฮฉ with a grid parameter โ„Ž such that ฮฉ={โˆช๐พโˆฃ๐พโˆˆ๐‘†โ„Ž} and โ„Ž=max๐พโˆˆ๐‘†โ„Žโ„Ž๐พ, where โ„Ž๐พ is the width of the subinterval ๐พ. Associated with the partition ๐‘†โ„Ž, we may define a finite-dimensional subspace ๐‘‰โ„ŽโŠ‚๐ป0๐œ‡/2(ฮฉ) with a basis {๐œ‘๐‘–}๐‘๐‘–=1 of piecewise polynomials. Then the semidiscrete variational problem is to find ๐‘ขโ„Žโˆˆ๐‘‰โ„Ž such that ๎€ท๐‘ขโ„Ž,๐‘ก๎€ธ=๎€ท๐œ…,๐‘ฃ๐œ‡โˆ‡๐œ‡๐‘ขโ„Ž๎€ธ+๎€ท๐‘“๎€ท๐‘ข,๐‘ฃโ„Ž๎€ธ๎€ธ,๐‘ฃ,โˆ€๐‘ฃโˆˆ๐‘‰โ„Ž,๐‘ข(3.1)โ„Ž(๐‘ฅ,0)=๐‘ข0,๐‘ข(3.2)โ„Ž(๐‘Ž,๐‘ก)=๐‘ขโ„Ž(๐‘,๐‘ก)=0.(3.3) Since ๐‘ขโ„Ž can be represented as ๐‘ขโ„Ž(๐‘ฅ,๐‘ก)=๐‘๎“๐‘–=1๐›ผ๐‘–(๐‘ก)๐œ‘๐‘–(๐‘ฅ),(3.4) we may rewrite (3.1) in a matrix form: ๐€ฬ‡๐ฎ(๐‘ก)+๐๐ฎ=๐…(๐ฎ),(3.5) where ๐‘ร—๐‘ matrices ๐€ and ๐ and vectors ๐ฎ and ๐… are ๎€ท๐‘Ž๐€=๐‘–๐‘—๎€ธ,๐‘Ž๐‘–๐‘—=๎€ท๐œ‘๐‘–,๐œ‘๐‘—๎€ธ,๎€ท๐‘๐=๐‘–๐‘—๎€ธ,๐‘๐‘–๐‘—๐œ…=โˆ’๐œ‡2๐ƒ๎€บ๎€ท๐œ‡/2๐œ‘๐‘–,๐ƒ(๐œ‡/2)โˆ—๐œ‘๐‘—๎€ธ+๎€ท๐ƒ๐œ‡/2๐œ‘๐‘—,๐ƒ(๐œ‡/2)โˆ—๐œ‘๐‘–,๎€ท๐น๎€ธ๎€ป๐…(๐ฎ)=๐‘—๎€ธ,๐น๐‘—=๎ƒฉ๐‘“๎ƒฉ๐‘๎“๐‘™=1๐›ผ๐‘™๐œ‘๐‘™๎ƒช,๐œ‘๐‘—๎ƒช,๎€ท๐›ผ๐ฎ=1(๐‘ก),๐›ผ2(๐‘ก),โ€ฆ,๐›ผ๐‘๎€ธ(๐‘ก)๐‘‡.(3.6)

It follows from โˆ‘๐‘๐‘–,๐‘—=1๐›ผ๐‘–๐›ผ๐‘—(๐œ‘๐‘–,๐œ‘๐‘—โˆ‘)=(๐‘๐‘–=1๐›ผ๐‘–๐œ‘๐‘–,โˆ‘๐‘๐‘—=1๐›ผ๐‘—๐œ‘๐‘—)โ‰ฅ0 and Lemma 2.4 that matrices ๐€ and ๐ are nonnegative definite and nonsingular. Thus this system (3.5) of ordinary differential equations has a unique solution since ๐‘“ is locally Lipschitz continuous.

The stability for the semidiscrete variational problem (3.1) can be obtained by following the proof of Theorem 2.5, which is โ€–โ€–๐‘ขโ„Žโ€–โ€–๐ฟ2(ฮฉ)โ€–โ€–๐‘ขโ‰ค๐ถ0โ€–โ€–๐ฟ2(ฮฉ).(3.7)

Now we will consider estimates of error between the weak solution of (2.13) and the one of semidiscrete form (3.1). The finite dimensional subspace ๐‘‰โ„ŽโŠ‚๐ป0๐œ‡/2(ฮฉ) is chosen so that the interpolation ๐ผโ„Ž๐‘ข of ๐‘ข satisfies an approximation property [9, 28]: for ๐‘ขโˆˆ๐ป๐›พ(ฮฉ), 0<๐›พโ‰ค๐‘›, and 0โ‰ค๐‘ โ‰ค๐›พ, there exists a constant ๐ถ depending only on ฮฉ such that โ€–โ€–๐‘ขโˆ’๐ผโ„Ž๐‘ขโ€–โ€–๐ป๐‘ (ฮฉ)โ‰ค๐ถโ„Ž๐›พโˆ’๐‘ โ€–๐‘ขโ€–๐ป๐›พ(ฮฉ).(3.8) Since the norm โ€–โ‹…โ€–๐ป๐‘ (ฮฉ) is equivalent to the seminorm |โ‹…|๐ป๐‘ (ฮฉ), we may replace (3.8) by the relation โ€–โ€–๐‘ขโˆ’๐ผโ„Ž๐‘ขโ€–โ€–๐ป๐‘ (ฮฉ)โ‰ค๐ถโ„Ž๐›พโˆ’๐‘ |๐‘ข|๐ป๐›พ(ฮฉ).(3.9)

Further we need an adjoint problem to find ๐‘คโˆˆ๐ป๐œ‡(ฮฉ)โˆฉ๐ป0๐œ‡/2(ฮฉ) satisfying โˆ’๐œ…๐œ‡โˆ‡๐œ‡๐‘ค=๐‘”,inฮฉ,๐‘ค=0,on๐œ•ฮฉ.(3.10) Bai and Lรผ [29] have proved existence of a solution to the problem (3.10). We assume as in Ervin and Roop [24] that the solution ๐‘ค satisfies the regularity โ€–๐‘คโ€–๐ป๐œ‡(ฮฉ)โ‰ค๐ถโ€–๐‘”โ€–๐ฟ2(ฮฉ)3,๐œ‡โ‰ 2,(3.11)โ€–๐‘คโ€–๐ป๐œ‡โˆ’๐œ–(ฮฉ)โ‰ค๐ถโ€–๐‘”โ€–๐ฟ2(ฮฉ)3,๐œ‡=21,0<๐œ–<2.(3.12)

Let ฬƒ๐‘ขโ„Ž=๐‘ƒโ„Ž๐‘ข be the elliptic projection ๐‘ƒโ„Žโˆถ๐ป0๐œ‡/2(ฮฉ)โ†’๐‘‰โ„Ž of the exact solution ๐‘ข, which is defined by โˆ’๐œ…๐œ‡๎€ทโˆ‡๐œ‡๎€ท๐‘ขโˆ’ฬƒ๐‘ขโ„Ž๎€ธ๎€ธ,๐‘ฃ=0,โˆ€๐‘ฃโˆˆ๐‘‰โ„Ž.(3.13) Let ๐œƒ=๐‘ขโ„Žโˆ’ฬƒ๐‘ขโ„Ž and ๐œŒ=ฬƒ๐‘ขโ„Žโˆ’๐‘ข. Then the error is expressed as ๐‘’โ„Ž=๐‘ขโ„Ž๎€ท๐‘ขโˆ’๐‘ข=โ„Žโˆ’ฬƒ๐‘ขโ„Ž๎€ธ+๎€ทฬƒ๐‘ขโ„Ž๎€ธโˆ’๐‘ข=๐œƒ+๐œŒ.(3.14)

First, we consider the following estimates on ๐œŒ.

Lemma 3.1. Let ฬƒ๐‘ขโ„Ž be a solution of (3.13) and let ๐‘ขโˆˆ๐ป๐œ‡(ฮฉ)โˆฉ๐ป0๐œ‡/2(ฮฉ) be the solution of (2.13). Let ๐œŒ(๐‘ก)=ฬƒ๐‘ขโ„Ž(๐‘ก)โˆ’๐‘ข(๐‘ก). Then there is a constant ๐ถ such that โ€–โ€–๐œŒ(๐‘ก)๐ฟ2(ฮฉ)โ€–โ‰ค๐ถโ„Žฬƒ๐›พโ€–๐‘ข(๐‘ก)๐ป๐›พ(ฮฉ),โ€–โ€–๐œŒ๐‘ก(โ€–โ€–๐‘ก)๐ฟ2(ฮฉ)(โ‰ค๐ถโ„Žฬƒ๐›พโ€–๐‘ข๐‘ก)โ€–๐ป๐›พ(ฮฉ),(3.15) where ฬƒ๐›พ=๐œ‡ if ๐œ‡โ‰ 3/2 and ฬƒ๐›พ=๐œ‡โˆ’๐œ–, 0<๐œ–<1/2 if ๐œ‡=3/2.

Proof. It follows from the fractional Poincarรฉ-Friedrichโ€™s inequality and the adjoint property (2.12) that for ๐œ“,๐œ’โˆˆ๐‘‰โ„ŽโŠ‚๐ป0๐œ‡/2(ฮฉ)(๐ƒ๐œ‡๎€œ๐œ“,๐œ’)=๐‘๐‘Ž๎€ท๐ƒ๐œ‡/2๐œ“๎€ธ๐ƒ(๐œ‡/2)โˆ—โ‰ค||๐œ“||๐œ’๐‘‘๐‘ฅ๐ฝ๐œ‡/2๐ฟ,0(ฮฉ)||๐œ’||๐ฝ๐œ‡/2๐‘…,0(ฮฉ)โ‰ค๐ถโ€–๐œ“โ€–๐ป0๐œ‡/2(ฮฉ)โ€–๐œ’โ€–๐ป0๐œ‡/2(ฮฉ).(3.16) Similarly we obtain ๎€ท๐ƒ๐œ‡โˆ—๎€ธ=๎€œ๐œ“,๐œ’๐‘๐‘Ž๎€ท๐ƒ(๐œ‡/2)โˆ—๐œ“๎€ธ๐ƒ๐œ‡/2๐œ’๐‘‘๐‘ฅโ‰ค๐ถโ€–๐œ“โ€–๐ป0๐œ‡/2(ฮฉ)โ€–๐œ’โ€–๐ป0๐œ‡/2(ฮฉ).(3.17)
It follows from Lemma 2.4 that for ๐‘ฃโˆˆ๐‘‰โ„Ž๐œ…๐œ‡|||๎‚€๐œ‡cos๐œ‹โ‹…2๎‚|||||๐‘ขโˆ’ฬƒ๐‘ขโ„Ž||2๐ป0๐œ‡/2(ฮฉ)โ‰คโˆ’๐œ…๐œ‡๎€ทโˆ‡๐œ‡๎€ท๐‘ขโˆ’ฬƒ๐‘ขโ„Ž๎€ธ,๐‘ขโˆ’ฬƒ๐‘ขโ„Ž๎€ธโ‰คโˆ’๐œ…๐œ‡๎€ทโˆ‡๐œ‡๎€ท๐‘ขโˆ’ฬƒ๐‘ขโ„Ž๎€ธ๎€ธ,๐‘ขโˆ’๐‘ฃโˆ’๐œ…๐œ‡๎€ทโˆ‡๐œ‡๎€ท๐‘ขโˆ’ฬƒ๐‘ขโ„Ž๎€ธ,๐‘ฃโˆ’ฬƒ๐‘ขโ„Ž๎€ธโ€–โ€–โ‰ค๐ถ๐‘ขโˆ’ฬƒ๐‘ขโ„Žโ€–โ€–๐ป0๐œ‡/2(ฮฉ)โ€–๐‘ขโˆ’๐‘ฃโ€–๐ป0๐œ‡/2(ฮฉ).(3.18) Using the equivalence of seminorms and norms, we obtain โ€–โ€–๐‘ขโˆ’ฬƒ๐‘ขโ„Žโ€–โ€–๐ป0๐œ‡/2(ฮฉ)โ‰ค๐ถinf๐‘ฃโˆˆ๐‘‰โ„Žโ€–๐‘ขโˆ’๐‘ฃโ€–๐ป0๐œ‡/2(ฮฉ)โ€–โ€–โ‰ค๐ถ๐‘ขโˆ’๐ผโ„Ž๐‘ขโ€–โ€–๐ป0๐œ‡/2(ฮฉ).(3.19)
In case of ๐œ‡โ‰ 3/2 and ๐‘ฃโˆˆ๐‘‰โ„Ž, by taking ๐‘”=๐œŒ in (3.10) and using (3.13), (3.16)-(3.17) and the adjoint property (2.12), we have (๐œŒ,๐œŒ)=โˆ’๐œ…๐œ‡(โˆ‡๐œ‡๐‘ค,๐œŒ)=โˆ’๐œ…๐œ‡(โˆ‡๐œ‡(๐‘คโˆ’๐‘ฃ),๐œŒ)โˆ’๐œ…๐œ‡(โˆ‡๐œ‡๐œŒ,๐‘ฃ)=โˆ’๐œ…๐œ‡(โˆ‡๐œ‡(๐‘คโˆ’๐‘ฃ),๐œŒ)โ‰ค๐ถโ€–๐‘คโˆ’๐‘ฃโ€–๐ป0๐œ‡/2(ฮฉ)โ€–๐œŒโ€–๐ป0๐œ‡/2(ฮฉ).(3.20) Taking ๐‘ฃ=๐ผโ„Ž๐‘ค in the previously mentioned inequalities, we have โ€–๐œŒโ€–2๐ฟ2(ฮฉ)โ€–โ€–โ‰ค๐ถ๐‘คโˆ’๐ผโ„Ž๐‘คโ€–โ€–๐ป0๐œ‡/2(ฮฉ)โ€–๐œŒโ€–๐ป0๐œ‡/2(ฮฉ)โ‰ค๐ถโ„Ž๐œ‡/2โ€–๐‘คโ€–๐ป๐œ‡(ฮฉ)โ€–โ€–๐‘ขโˆ’๐ผโ„Ž๐‘ขโ€–โ€–๐ป0๐œ‡/2(ฮฉ)โ‰ค๐ถโ„Ž๐œ‡/2โ€–๐œŒโ€–๐ฟ2(ฮฉ)โ„Ž๐œ‡/2โ€–๐‘ขโ€–๐ป๐œ‡(ฮฉ).(3.21) Thus we obtain โ€–๐œŒโ€–๐ฟ2(ฮฉ)โ‰ค๐ถโ„Ž๐œ‡โ€–๐‘ขโ€–๐ป๐œ‡(ฮฉ).(3.22) We now differentiate (3.13). Then we obtain โˆ’๐œ…๐œ‡(โˆ‡๐œ‡๐œŒ๐‘ก,๐‘ฃ)=0 for all ๐‘ฃโˆˆ๐‘‰โ„Ž. Using the previous duality arguments again, we have โ€–โ€–๐œŒ๐‘กโ€–โ€–๐ฟ2(ฮฉ)โ‰ค๐ถโ„Ž๐œ‡โ€–๐‘ขโ€–๐ป๐œ‡(ฮฉ).(3.23)
In case of ๐œ‡=3/2, we can similarly prove (3.15) by applying the assumption (3.12). This completes the proof.

We now consider the estimates on ๐œƒ.

Lemma 3.2. Let ๐‘ขโ„Ž and ฬƒ๐‘ขโ„Ž be the solutions of (3.1)โ€“(3.3) and (3.13), respectively. Let ๐œƒ(๐‘ก)=๐‘ขโ„Ž(๐‘ก)โˆ’ฬƒ๐‘ขโ„Ž(๐‘ก). Then there is a constant ๐ถ such that โ€–โ€–๐œƒ(๐‘ก)๐ฟ2(ฮฉ),โ‰ค๐ถโ„Žฬƒ๐›พ(3.24) where ฬƒ๐›พ=๐œ‡ if ๐œ‡โ‰ 3/2 and ฬƒ๐›พ=๐œ‡โˆ’๐œ–, 0<๐œ–<1/2 if ๐œ‡=3/2.

Proof. It follows from (3.1) and (3.13) that for ๐‘ฃโˆˆ๐‘‰โ„Ž, ๎€ท๐œƒ๐‘ก๎€ธ,๐‘ฃโˆ’๐œ…๐œ‡(โˆ‡๐œ‡๎€ท๐‘“๎€ท๐‘ข๐œƒ,๐‘ฃ)=โ„Ž๎€ธ๎€ธโˆ’๎€ท๐œŒโˆ’๐‘“(๐‘ข),๐‘ฃ๐‘ก๎€ธ.,๐‘ฃ(3.25) Replacing ๐‘ฃ=๐œƒ in (3.25), we obtain 12๐‘‘๐‘‘๐‘กโ€–๐œƒโ€–2๐ฟ2(ฮฉ)โ‰ค๐ถ๐‘™โ€–โ€–๐‘ขโ„Žโ€–โ€–โˆ’๐‘ข๐ฟ2(ฮฉ)โ€–๐œƒโ€–๐ฟ2(ฮฉ)+โ€–โ€–๐œŒ๐‘กโ€–โ€–๐ฟ2(ฮฉ)โ€–๐œƒโ€–๐ฟ2(ฮฉ).(3.26) Using Young's inequality ๐‘‘๐‘‘๐‘กโ€–๐œƒโ€–2๐ฟ2(ฮฉ)๎€ทโ€–โ€–๐‘ขโ‰ค๐ถโ„Žโˆ’ฬƒ๐‘ขโ„Žโ€–โ€–๐ฟ2(ฮฉ)+โ€–โ€–ฬƒ๐‘ขโ„Žโ€–โ€–โˆ’๐‘ข๐ฟ2(ฮฉ)๎€ธโ€–๐œƒโ€–๐ฟ2(ฮฉ)+โ€–โ€–๐œŒ๐‘กโ€–โ€–๐ฟ2(ฮฉ)โ€–๐œƒโ€–๐ฟ2(ฮฉ)๎€ทโ‰ค๐ถโ€–๐œƒโ€–๐ฟ2(ฮฉ)+โ€–๐œŒโ€–๐ฟ2(ฮฉ)+โ€–โ€–๐œŒ๐‘กโ€–โ€–๐ฟ2(ฮฉ)๎€ธโ€–๐œƒโ€–๐ฟ2(ฮฉ)โ‰ค๐ถ1โ€–๐œƒโ€–2๐ฟ2(ฮฉ)+๐ถ2โ€–๐œŒโ€–2๐ฟ2(ฮฉ)+๐ถ3โ€–โ€–๐œŒ๐‘กโ€–โ€–2๐ฟ2(ฮฉ).(3.27) Integration on time ๐‘ก gives โ€–๐œƒ(๐‘ก)โ€–2๐ฟ2(ฮฉ)โ‰คโ€–๐œƒ(0)โ€–2๐ฟ2(ฮฉ)๎€œ+๐ถ๐‘ก0โ€–๐œƒโ€–2๐ฟ2(ฮฉ)๎€œ๐‘‘๐‘ +๐ถ๐‘ก0๎‚€โ€–๐œŒโ€–2๐ฟ2(ฮฉ)+โ€–โ€–๐œŒ๐‘กโ€–โ€–2๐ฟ2(ฮฉ)๎‚๐‘‘๐‘ .(3.28) Applying Gronwall's inequality, we obtain โ€–๐œƒ(๐‘ก)โ€–2๐ฟ2(ฮฉ)โ‰ค๐ถ1โ€–๐œƒ(0)โ€–2๐ฟ2(ฮฉ)+๐ถ2๎€œ๐‘ก0๎‚€โ€–๐œŒโ€–2๐ฟ2(ฮฉ)+โ€–โ€–๐œŒ๐‘กโ€–โ€–2๐ฟ2(ฮฉ)๎‚๐‘‘๐‘ .(3.29) Since โ€–๐œƒ(0)โ€–๐ฟ2(ฮฉ)โ‰คโ€–โ€–๐‘ขโ„Žโ€–โ€–(0)โˆ’๐‘ข(0)๐ฟ2(ฮฉ)+โ€–โ€–ฬƒ๐‘ขโ„Žโ€–โ€–(0)โˆ’๐‘ข(0)๐ฟ2(ฮฉ)โ€–โ€–๐‘ขโ‰ค๐ถโ„Žฬƒ๐›พ0โ€–โ€–๐ป๐›พ(ฮฉ),(3.30) we obtain the desired inequality โ€–๐œƒ(๐‘ก)โ€–๐ฟ2(ฮฉ),โ‰ค๐ถโ„Žฬƒ๐›พ(3.31) where ฬƒ๐›พ=๐œ‡ if ๐œ‡โ‰ 3/2 and ฬƒ๐›พ=๐œ‡โˆ’๐œ–, 0<๐œ–<1/2, if ๐œ‡=3/2.

Combining Lemmas 3.1 and 3.2, we obtain the following error estimates.

Theorem 3.3. Let ๐‘ขโ„Ž and ๐‘ข be the solutions of (3.1)โ€“(3.3) and (1.2)โ€“(1.4), respectively. Then there is a constant ๐ถ(๐‘ข) such that โ€–โ€–๐‘ข(๐‘ก)โˆ’๐‘ขโ„Žโ€–โ€–(๐‘ก)๐ฟ2(ฮฉ)โ‰ค๐ถ(๐‘ข)โ„Ž๐œ‡3,๐œ‡โ‰ 2,โ€–โ€–๐‘ข(๐‘ก)โˆ’๐‘ขโ„Žโ€–โ€–(๐‘ก)๐ฟ2(ฮฉ)โ‰ค๐ถ(๐‘ข)โ„Ž๐œ‡โˆ’๐œ–3,๐œ‡=21,0<๐œ–<2.(3.32)

4. The Fully Discrete Variational Form

In this section, we consider a fully discrete variational formulation of (1.2). Existence and uniqueness of numerical solutions for the fully discrete variational formulation are discussed. The corresponding error estimates are also analyzed.

For the temporal discretization let ๐‘˜=๐‘‡/๐‘€ for a positive integer ๐‘€ and ๐‘ก๐‘š=๐‘š๐‘˜. Let ๐‘ข๐‘š be the solution of the backward Euler method defined by ๐‘ข๐‘š+1โˆ’๐‘ข๐‘š๐‘˜=๐œ…๐œ‡โˆ‡๐œ‡๐‘ข๐‘š+1๎€ท๐‘ข+๐‘“๐‘š+1๎€ธ(4.1) with an initial condition ๐‘ข0(๐‘ฅ)=๐‘ข0,๐‘ฅโˆˆฮฉ=(๐‘Ž,๐‘)(4.2) and boundary conditions ๐‘ข๐‘š+1(๐‘Ž)=๐‘ข๐‘š+1(๐‘)=0,๐‘š=0,1,โ€ฆ,๐‘€โˆ’1.(4.3) Then we get the fully discrete variational formulation of (1.2) to find ๐‘ข๐‘š+1โˆˆ๐ป0๐œ‡/2(ฮฉ) such that for all ๐‘ฃโˆˆ๐ป0๐œ‡/2(ฮฉ)๎€ท๐‘ข๐‘š+1๎€ธ๎€ท๐œ…,๐‘ฃโˆ’๐‘˜๐œ‡โˆ‡๐œ‡๐‘ข๐‘š+1๎€ธ=๎€ท๎€ท๐‘ข,๐‘ฃ๐‘˜๐‘“๐‘š+1๎€ธ๎€ธ,๐‘ฃ+(๐‘ข๐‘š,๐‘ฃ).(4.4) Thus a finite Galerkin solution ๐‘ขโ„Ž๐‘š+1โˆˆ๐‘‰โ„ŽโŠ‚๐ป0๐œ‡/2(ฮฉ) is a solution of the equation ๎€ท๐‘ขโ„Ž๐‘š+1,๐‘ฃโ„Ž๎€ธโˆ’๐‘˜๐œ…๐œ‡๎€ทโˆ‡๐œ‡๐‘ขโ„Ž๐‘š+1,๐‘ฃโ„Ž๎€ธ๎€ท๐‘“๎€ท๐‘ข=๐‘˜โ„Ž๐‘š+1๎€ธ,๐‘ฃโ„Ž๎€ธ+๎€ท๐‘ข๐‘šโ„Ž,๐‘ฃโ„Ž๎€ธ,โˆ€๐‘ฃโ„Žโˆˆ๐‘‰โ„Ž(4.5) with an initial condition ๐‘ข0โ„Ž=๐‘ข0(4.6) and boundary conditions ๐‘ขโ„Ž๐‘š+1(๐‘Ž)=๐‘ขโ„Ž๐‘š+1(๐‘)=0,๐‘š=0,1,โ€ฆ,๐‘€โˆ’1.(4.7)

Now we prove the existence and uniqueness of solutions for (4.5) using the Brouwer fixed-point theorem.

Theorem 4.1. There exists a unique solution ๐‘ขโ„Ž๐‘š+1โˆˆ๐‘‰โ„ŽโŠ‚๐ป0๐œ‡/2(ฮฉ) of (4.5)โ€“(4.7).

Proof. Let ๐บ๎€ท๐‘ขโ„Ž๐‘š+1๎€ธ=๐‘ขโ„Ž๐‘š+1โˆ’๐‘˜๐œ…๐œ‡โˆ‡๐œ‡๐‘ขโ„Ž๐‘š+1๎€ท๐‘ขโˆ’๐‘˜๐‘“โ„Ž๐‘š+1๎€ธโˆ’๐‘ข๐‘šโ„Ž.(4.8) Then ๐บ(๐‘ฃ) is obviously a continuous function from ๐‘‰โ„Ž to ๐‘‰โ„Ž. In order to show the existence of solution for ๐บ(๐‘ฃ)=0, we adopt the mathematical induction. Assume that ๐‘ข0โ„Ž,๐‘ข1โ„Ž,โ€ฆ,๐‘ข๐‘šโ„Ž exist for ๐‘š<๐‘€. It follows from (1.8), Lemma 2.4, and Young's inequality that ๎€ท๐‘ข(๐บ(๐‘ฃ),๐‘ฃ)=(๐‘ฃ,๐‘ฃ)โˆ’๐‘šโ„Ž๎€ธ๎€ท๐œ…,๐‘ฃโˆ’๐‘˜๐œ‡โˆ‡๐œ‡๎€ธ๐‘ฃ,๐‘ฃโˆ’๐‘˜(๐‘“(๐‘ฃ),๐‘ฃ)โ‰ฅโ€–๐‘ฃโ€–2๐ฟ2(ฮฉ)โˆ’โ€–โ€–๐‘ข๐‘šโ„Žโ€–โ€–๐ฟ2(ฮฉ)โ€–๐‘ฃโ€–๐ฟ2(ฮฉ)+๐‘˜๐œ…๐œ‡|||๎‚€๐œ‡cos๐œ‹โ‹…2๎‚||||๐‘ฃ|2๐ป0๐œ‡/2(ฮฉ)โˆ’๐ถ๐‘“๐‘˜โ€–๐‘ฃโ€–2๐ฟ2(ฮฉ)โ‰ฅโ€–๐‘ฃโ€–2๐ฟ2(ฮฉ)โˆ’โ€–โ€–๐‘ข๐‘šโ„Žโ€–โ€–๐ฟ2(ฮฉ)โ€–๐‘ฃโ€–๐ฟ2(ฮฉ)โˆ’๐ถ๐‘“๐‘˜โ€–๐‘ฃโ€–2๐ฟ2(ฮฉ)โ‰ฅโ€–๐‘ฃโ€–2๐ฟ2(ฮฉ)โˆ’12๎‚€โ€–โ€–๐‘ข๐‘šโ„Žโ€–โ€–2๐ฟ2(ฮฉ)+โ€–๐‘ฃโ€–2๐ฟ2(ฮฉ)๎‚โˆ’๐ถ๐‘“๐‘˜โ€–๐‘ฃโ€–2๐ฟ2(ฮฉ)=๎‚€12โˆ’๐ถ๐‘“๐‘˜๎‚โ€–๐‘ฃโ€–2๐ฟ2(ฮฉ)โˆ’12โ€–โ€–๐‘ข๐‘šโ„Žโ€–โ€–2๐ฟ2(ฮฉ).(4.9) If we take sufficiently small ๐‘˜ so that ๐‘˜<1/2๐ถ๐‘“ and โ€–๐‘ฃโ€–๐ฟ2(ฮฉ)>โ€–๐‘ข๐‘šโ„Žโ€–๐ฟ2(ฮฉ)/(1โˆ’2๐ถ๐‘“๐‘˜), then the Brouwer's fixed-point theorem implies the existence of a solution.
For the proof of the uniqueness of solutions, we assume that ๐‘ข and ๐‘ฃ are two solutions of (4.5). Then we obtain (๐‘ขโˆ’๐‘ฃ,๐œ“)=๐‘˜๐œ…๐œ‡(โˆ‡๐œ‡(๐‘ขโˆ’๐‘ฃ),๐œ“)+๐‘˜(๐‘“(๐‘ข)โˆ’๐‘“(๐‘ฃ),๐œ“),โˆ€๐œ“โˆˆ๐‘‰โ„ŽโŠ‚๐ป0๐œ‡/2(ฮฉ).(4.10) Replacing ๐œ“=๐‘ขโˆ’๐‘ฃ in the above equation and applying Lemma 2.4, we obtain โ€–๐‘ขโˆ’๐‘ฃโ€–2๐ฟ2(ฮฉ)โ‰คโˆ’๐‘˜๐œ…๐œ‡|||๎‚€๐œ‡cos๐œ‹โ‹…2๎‚||||๐‘ขโˆ’๐‘ฃ|๐ป0๐œ‡/2(ฮฉ)+๐‘˜โ€–๐‘“(๐‘ข)โˆ’๐‘“(๐‘ฃ)โ€–๐ฟ2(ฮฉ)โ€–๐‘ขโˆ’๐‘ฃโ€–๐ฟ2(ฮฉ)โ‰ค๐‘˜โ€–๐‘“(๐‘ข)โˆ’๐‘“(๐‘ฃ)โ€–๐ฟ2(ฮฉ)โ€–๐‘ขโˆ’๐‘ฃโ€–๐ฟ2(ฮฉ)โ‰ค๐‘˜๐ถ๐‘™โ€–๐‘ขโˆ’๐‘ฃโ€–2๐ฟ2(ฮฉ).(4.11) This implies ๐‘ขโˆ’๐‘ฃ=0 since ๐‘ข(0)=๐‘ฃ(0).

The following theorem presents the unconditional stability for (4.4).

Theorem 4.2. The fully discrete scheme (4.4) is unconditionally stable. In fact, for any ๐‘šโ€–โ€–๐‘ข๐‘š+1โ€–โ€–๐ฟ2(ฮฉ)โ€–โ€–๐‘ขโ‰ค๐ถ0โ€–โ€–๐ฟ2(ฮฉ).(4.12)

Proof. It follows from (1.8), Lemma 2.4, and Young's inequality that by taking ๐‘ฃ=๐‘ข๐‘š+1 in (4.4), we obtain ๎€ท๐‘ข0=๐‘š+1,๐‘ข๐‘š+1๎€ธ๎€ท๐œ…โˆ’๐‘˜๐œ‡โˆ‡๐œ‡๐‘ข๐‘š+1,๐‘ข๐‘š+1๎€ธ๎€ท๐‘“๎€ท๐‘ขโˆ’๐‘˜๐‘š+1๎€ธ,๐‘ข๐‘š+1๎€ธโˆ’๎€ท๐‘ข๐‘š,๐‘ข๐‘š+1๎€ธโ‰ฅโ€–โ€–๐‘ข๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ)+๐‘˜๐œ…๐œ‡|||๎‚€๐œ‡cos๐œ‹โ‹…2๎‚|||||๐‘ข๐‘š+1||2๐ป0๐œ‡/2(ฮฉ)โˆ’๐ถ๐‘“๐‘˜โ€–โ€–๐‘ข๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ)โˆ’โ€–๐‘ข๐‘šโ€–๐ฟ2(ฮฉ)โ€–โ€–๐‘ข๐‘š+1โ€–โ€–๐ฟ2(ฮฉ)โ‰ฅ12โ€–โ€–๐‘ข๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ)+๐‘˜๐œ…๐œ‡|||๎‚€๐œ‡cos๐œ‹โ‹…2๎‚|||||๐‘ข๐‘š+1||2๐ป0๐œ‡/2(ฮฉ)โˆ’๐ถ๐‘“๐‘˜โ€–โ€–๐‘ข๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ)โˆ’12โ€–๐‘ข๐‘šโ€–2๐ฟ2(ฮฉ).(4.13) Then 12โ€–โ€–๐‘ข๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ)โ‰ค12โ€–โ€–๐‘ข๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ)+๐‘˜๐œ…๐œ‡|||๎‚€๐œ‡cos๐œ‹โ‹…2๎‚|||||๐‘ข๐‘š+1||2๐ป0๐œ‡/2(ฮฉ)โ‰ค๐ถ๐‘“๐‘˜โ€–โ€–๐‘ข๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ)+12โ€–๐‘ข๐‘šโ€–2๐ฟ2(ฮฉ).(4.14)
Adding the above inequality from ๐‘š=0 to ๐‘š, we obtain ๎€ท1โˆ’2๐ถ๐‘“๐‘˜๎€ธโ€–โ€–๐‘ข๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ)โ‰คโ€–โ€–๐‘ข0โ€–โ€–2๐ฟ2(ฮฉ)+2๐ถ๐‘“๐‘˜๐‘š๎“๐‘—=1โ€–โ€–๐‘ข๐‘—โ€–โ€–2๐ฟ2(ฮฉ).(4.15) Applying the discrete Gronwall's inequality with sufficiently small ๐‘˜ such that ๐‘˜<1/2๐ถ๐‘“, we obtain the desired result.

The following theorem is an error estimate for the fully discrete problem (4.4).

Theorem 4.3. Let ๐‘ข be the exact solution of (1.2) and let ๐‘ข๐‘š be the solution of (4.4). Then there is a constant ๐ถ such that โ€–โ€–๐‘ข๎€ท๐‘ก๐‘š๎€ธโˆ’๐‘ข๐‘šโ€–โ€–๐ฟ2(ฮฉ)โ‰ค๐ถ๐‘˜.(4.16)

Proof. Let ๐‘’๐‘š=๐‘ข(๐‘ก๐‘š)โˆ’๐‘ข๐‘š be the error at ๐‘ก๐‘š. It follows from (1.2) and (4.4) that for any ๐‘ฃโˆˆ๐ป0๐œ‡/2(ฮฉ)๎€ท๐‘’๐‘š+1๎€ธ๎€ท๐œ…,๐‘ฃโˆ’๐‘˜๐œ‡โˆ‡๐œ‡๐‘’๐‘š+1๎€ธ๎€ท๐‘“๎€ท๐‘ข๎€ท๐‘ก,๐‘ฃ=๐‘˜๐‘š+1๎€ท๐‘ข๎€ธ๎€ธโˆ’๐‘“๐‘š+1๎€ธ๎€ธ,๐‘ฃ+(๐‘’๐‘š๎€ท,๐‘ฃ)+๐‘˜๐‘Ÿ๐‘š+1๎€ธ,,๐‘ฃ(4.17) where ๐‘Ÿ=๐‘‚(๐‘˜). Taking ๐‘ฃ=๐‘’๐‘š+1, โ€–โ€–๐‘’๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ)โ‰คโ€–โ€–๐‘’๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ)+๐‘˜๐œ…๐œ‡|||๎‚€๐œ‡cos๐œ‹โ‹…2๎‚|||||๐‘’๐‘š+1||2๐ป0๐œ‡/2(ฮฉ)โ€–โ€–โ‰ค๐‘˜๐‘“(๐‘ข(๐‘ก๐‘š+1))โˆ’๐‘“(๐‘ข๐‘š+1)โ€–โ€–๐ฟ2(ฮฉ)โ€–โ€–๐‘’๐‘š+1โ€–โ€–๐ฟ2(ฮฉ)+โ€–๐‘’๐‘šโ€–๐ฟ2(ฮฉ)โ€–โ€–๐‘’๐‘š+1โ€–โ€–๐ฟ2(ฮฉ)+โ€–โ€–๐‘˜๐‘Ÿ๐‘š+1โ€–โ€–๐ฟ2(ฮฉ)โ€–โ€–๐‘’๐‘š+1โ€–โ€–๐ฟ2(ฮฉ).(4.18) Applying the locally Lipschitz continuity of ๐‘“ and Young's inequality, we obtain โ€–โ€–๐‘’๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ)โ‰ค๐‘˜๐ถ๐‘™โ€–โ€–๐‘’๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ)+โ€–๐‘’๐‘šโ€–๐ฟ2(ฮฉ)โ€–โ€–๐‘’๐‘š+1โ€–โ€–๐ฟ2(ฮฉ)+โ€–โ€–๐‘˜๐‘Ÿ๐‘š+1โ€–โ€–๐ฟ2(ฮฉ)โ€–โ€–๐‘’๐‘š+1โ€–โ€–๐ฟ2(ฮฉ)โ‰ค๐‘˜๐ถ๐‘™โ€–โ€–๐‘’๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ)+๐œ€1โ€–๐‘’๐‘šโ€–2๐ฟ2(ฮฉ)+14๐œ€1โ€–โ€–๐‘’๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ)+๐œ€2โ€–โ€–๐‘˜๐‘Ÿ๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ)+14๐œ€2โ€–โ€–๐‘’๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ).(4.19) That is, ๎‚ต11โˆ’4๐œ€1โˆ’14๐œ€2๎‚ถโ€–โ€–๐‘’๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ)โ‰ค๐‘˜๐ถ๐‘™โ€–โ€–๐‘’๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ)+๐œ€1โ€–๐‘’๐‘šโ€–2๐ฟ2(ฮฉ)+๐œ€2โ€–โ€–๐‘˜๐‘Ÿ๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ).(4.20) Denoting ๐œ€0=1โˆ’1/4๐œ€1โˆ’1/4๐œ€2 and adding the above equation from ๐‘š=0 to ๐‘š, we obtain ๎€ท๐œ€0โˆ’๐‘˜๐ถ๐‘™๎€ธโ€–โ€–๐‘’๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ)โ‰ค๐œ€1โ€–โ€–๐‘’0โ€–โ€–2๐ฟ2(ฮฉ)+๎€ท๐‘˜๐ถ๐‘™+๐œ€1โˆ’๐œ€0๎€ธ๐‘š๎“๐‘–=1โ€–โ€–๐‘’๐‘–โ€–โ€–2๐ฟ2(ฮฉ)+๐œ€2๐‘š+1๎“๐‘–=1โ€–โ€–๐‘˜๐‘Ÿ๐‘–โ€–โ€–2๐ฟ2(ฮฉ).(4.21) Applying the discrete Gronwall's inequality with sufficiently small ๐‘˜ such that (๐œ€0โˆ’๐œ€1)/๐ถ๐‘™<๐‘˜<๐œ€0/๐ถ๐‘™, we obtain the desired result since โˆ‘๐‘š+1๐‘–=1โ€–๐‘˜๐‘Ÿ๐‘–โ€–๐ฟ2(ฮฉ)โ‰ค๐ถ๐‘˜ and โ€–๐‘’0โ€–๐ฟ2(ฮฉ)=โ€–๐‘ข(0)โˆ’๐‘ข0โ€–๐ฟ2(ฮฉ)=0.

As in the previous section, denote ๐œƒ๐‘š+1=๐‘ขโ„Ž๐‘š+1โˆ’ฬƒ๐‘ขโ„Ž๐‘š+1 and ๐œŒ๐‘š+1=ฬƒ๐‘ขโ„Ž๐‘š+1โˆ’๐‘ข(๐‘ก๐‘š+1). Here ฬƒ๐‘ขโ„Ž๐‘š+1 is the elliptic projection of ๐‘ข(๐‘ก๐‘š+1) defined in (3.13). Then ๐‘’โ„Ž๐‘š+1=๐œƒ๐‘š+1+๐œŒ๐‘š+1.(4.22)

Theorem 4.4. Let ๐‘ข be the exact solution of (1.2)โ€“(1.4) and let {๐‘ข๐‘šโ„Ž}๐‘€๐‘š=0 be the solution of (4.5)โ€“(4.7). Then when ๐œ‡โ‰ 3/2โ€–โ€–๐‘ข(๐‘ก๐‘š+1)โˆ’๐‘ขโ„Ž๐‘š+1โ€–โ€–๐ฟ2(ฮฉ)โ‰ค๐ถ๐‘˜+๐ถโ„Ž๐œ‡โ€–โ€–๐‘ข๎€ท๐‘ก๐‘š+1๎€ธโ€–โ€–๐ป๐œ‡(ฮฉ)(4.23) and when ๐œ‡=3/2,0<๐œ–<1/2, โ€–โ€–๐‘ข(๐‘ก๐‘š+1)โˆ’๐‘ขโ„Ž๐‘š+1โ€–โ€–๐ฟ2(ฮฉ)โ‰ค๐ถ๐‘˜+๐ถโ„Ž๐œ‡โˆ’๐œ–โ€–โ€–๐‘ข(๐‘ก๐‘š+1)โ€–โ€–๐ป๐œ‡โˆ’๐œ–(ฮฉ).(4.24)

Proof. Since we know the estimates on ๐œŒ from Lemma 3.1, we have only to show boundedness of ๐œƒ๐‘š+1. Using the property (3.13), we obtain for ๐‘ฃโˆˆ๐‘‰โ„Ž๎€ท๐œƒ๐‘š+1๎€ธ๎€ท๐œ…,๐‘ฃโˆ’๐‘˜๐œ‡โˆ‡๐œ‡๐œƒ๐‘š+1๎€ธ๎€ท๐‘“๎€ท๐‘ข,๐‘ฃ=๐‘˜โ„Ž๐‘š+1๎€ธ๎€ท๐‘ข๎€ท๐‘กโˆ’๐‘“๐‘š+1๎€ธ+๎€ท๐‘ข๎€ธ๎€ธ,๐‘ฃ๐‘šโ„Ž๎€ท๐‘กโˆ’๐‘ข๐‘š๎€ธ๎€ธโˆ’๎€ท,๐‘ฃ๐‘˜๐‘Ÿ๐‘š+1๎€ธโˆ’๎€ท๐œŒ,๐‘ฃ๐‘š+1๎€ธ,,๐‘ฃ(4.25) where ๐‘Ÿ=๐‘‚(๐‘˜).
Taking ๐‘ฃ=๐œƒ๐‘š+1 and applying Lemma 2.4, the locally Lipschitz continuity of ๐‘“, Young's inequality, and the triangle inequality, we obtain โ€–โ€–๐œƒ๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ)โ‰คโ€–โ€–๐œƒ๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ)+๐‘˜๐œ…๐œ‡|||๎‚€๐œ‡cos๐œ‹โ‹…2๎‚|||||๐œƒ๐‘š+1||2๐ป0๐œ‡/2(ฮฉ)โ€–โ€–โ‰ค๐‘˜๐‘“(๐‘ขโ„Ž๐‘š+1)โˆ’๐‘“(๐‘ข(๐‘ก๐‘š+1โ€–โ€–))๐ฟ2(ฮฉ)โ€–โ€–๐œƒ๐‘š+1โ€–โ€–๐ฟ2(ฮฉ)+โ€–โ€–๐‘’๐‘šโ„Žโ€–โ€–๐ฟ2(ฮฉ)โ€–โ€–๐œƒ๐‘š+1โ€–โ€–๐ฟ2(ฮฉ)+โ€–โ€–๐‘˜๐‘Ÿ๐‘š+1โ€–โ€–๐ฟ2(ฮฉ)โ€–โ€–๐œƒ๐‘š+1โ€–โ€–๐ฟ2(ฮฉ)+โ€–โ€–๐œŒ๐‘š+1โ€–โ€–๐ฟ2(ฮฉ)โ€–โ€–๐œƒ๐‘š+1โ€–โ€–๐ฟ2(ฮฉ)โ‰ค๐‘˜๐ถ๐‘™โ€–โ€–๐‘’โ„Ž๐‘š+1โ€–โ€–๐ฟ2(ฮฉ)โ€–โ€–๐œƒ๐‘š+1โ€–โ€–๐ฟ2(ฮฉ)+๎€ทโ€–๐œƒ๐‘šโ€–๐ฟ2(ฮฉ)+โ€–๐œŒ๐‘šโ€–๐ฟ2(ฮฉ)๎€ธโ€–โ€–๐œƒ๐‘š+1โ€–โ€–๐ฟ2(ฮฉ)+โ€–โ€–๐‘˜๐‘Ÿ๐‘š+1โ€–โ€–๐ฟ2(ฮฉ)โ€–โ€–๐œƒ๐‘š+1โ€–โ€–๐ฟ2(ฮฉ)+โ€–โ€–๐œŒ๐‘š+1โ€–โ€–๐ฟ2(ฮฉ)โ€–โ€–๐œƒ๐‘š+1โ€–โ€–๐ฟ2(ฮฉ)โ‰ค๐‘˜๐ถ๐‘™๎‚ต11+4๐œ€6๎‚ถโ€–โ€–๐œƒ๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ)+๎‚ต14๐œ€3+14๐œ€4+14๐œ€5+14๐œ€6๎‚ถโ€–โ€–๐œƒ๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ)+๐œ€3โ€–๐œƒ๐‘šโ€–2๐ฟ2(ฮฉ)+๐œ€4โ€–โ€–๐‘˜๐‘Ÿ๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ)+๐œ€5โ€–๐œŒ๐‘šโ€–2๐ฟ2(ฮฉ)+๎€ท1+๐‘˜๐ถ๐‘™๎€ธ๐œ€6โ€–โ€–๐œŒ๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ).(4.26) This implies that ๎‚ต11โˆ’4๐œ€3โˆ’14๐œ€4โˆ’14๐œ€5โˆ’14๐œ€6๎‚ถโ€–โ€–๐œƒ๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ)โ‰ค๐‘˜๐ถ๐‘™๎‚ต11+4๐œ€6๎‚ถโ€–โ€–๐œƒ๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ)+๐œ€3โ€–๐œƒ๐‘šโ€–2๐ฟ2(ฮฉ)+๐œ€4โ€–โ€–๐‘˜๐‘Ÿ๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ)+๐œ€5โ€–๐œŒ๐‘šโ€–2๐ฟ2(ฮฉ)+๎€ท1+๐‘˜๐ถ๐‘™๎€ธ๐œ€6โ€–โ€–๐œŒ๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ).(4.27) Denote ๐œ€7=1โˆ’1/4๐œ€3โˆ’1/4๐œ€4โˆ’1/4๐œ€5โˆ’1/4๐œ€6 and ๐œ€8=1+1/4๐œ€6. Then adding the above inequality from ๐‘š=0 to ๐‘š, we obtain ๎€ท๐œ€7โˆ’๐‘˜๐ถ๐‘™๐œ€8๎€ธโ€–โ€–๐œƒ๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ)โ‰ค๐œ€3โ€–โ€–๐œƒ0โ€–โ€–2๐ฟ2(ฮฉ)+๎€ท๐‘˜๐ถ๐‘™๐œ€8+๐œ€3โˆ’๐œ€7๎€ธ๐‘š๎“๐‘–=1โ€–โ€–๐œƒ๐‘–โ€–โ€–2๐ฟ2(ฮฉ)+๐œ€4๐‘š+1๎“๐‘–=1โ€–โ€–๐‘˜๐‘Ÿ๐‘–โ€–โ€–2๐ฟ2(ฮฉ)+๐œ€5๐‘š๎“๐‘–=0โ€–โ€–๐œŒ๐‘–โ€–โ€–2๐ฟ2(ฮฉ)+๎€ท1+๐‘˜๐ถ๐‘™๎€ธ๐œ€6๐‘š+1๎“๐‘–=1โ€–โ€–๐œŒ๐‘–โ€–โ€–2๐ฟ2(ฮฉ).(4.28) Applying the discrete Gronwall's inequality with sufficiently small ๐‘˜ such that (๐œ€7โˆ’๐œ€3)/๐œ€8๐ถ๐‘™<๐‘˜<๐œ€7/๐ถ๐‘™๐œ€8, โ€–โ€–๐œƒ๐‘š+1โ€–โ€–2๐ฟ2(ฮฉ)โ‰ค๐ถ1โ€–โ€–๐œƒ0โ€–โ€–2๐ฟ2(ฮฉ)+๐ถ2๐‘š+1๎“๐‘–=1โ€–โ€–๐‘˜๐‘Ÿ๐‘–โ€–โ€–2๐ฟ2(ฮฉ)+๐ถ3๐‘š+1๎“๐‘–=0โ€–โ€–๐œŒ๐‘–โ€–โ€–2๐ฟ2(ฮฉ).(4.29) Also, using Lemma 3.1 and the initial conditions (1.3) and (4.6), we obtain โ€–โ€–๐œƒ0โ€–โ€–๐ฟ2(ฮฉ)โ‰คโ€–โ€–๐‘ข0โ„Žโ€–โ€–โˆ’๐‘ข(0)๐ฟ2(ฮฉ)+โ€–โ€–ฬƒ๐‘ข0โ„Žโ€–โ€–โˆ’๐‘ข(0)๐ฟ2(ฮฉ)โ€–โ€–๐‘ขโ‰ค๐ถโ„Žฬƒ๐›พ0โ€–โ€–๐ป๐›พ(ฮฉ).(4.30) Since โˆ‘๐‘š+1๐‘–=1โ€–๐‘˜๐‘Ÿ๐‘–โ€–๐ฟ2(ฮฉ)โ‰ค๐ถ๐‘˜, we get โ€–โ€–๐œƒ๐‘š+1โ€–โ€–๐ฟ2(ฮฉ),โ‰ค๐ถ๐‘˜+๐ถ(๐‘ข)โ„Žฬƒ๐›พ(4.31) where ฬƒ๐›พ=๐œ‡ if ๐œ‡โ‰ 3/2 and ฬƒ๐›พ=๐œ‡โˆ’๐œ–, 0<๐œ–<1/2, if ๐œ‡=3/2. Thus we obtain the desired result.

5. Numerical Experiments

In this section, we present numerical results for the Galerkin approximations which supports the theoretical analysis discussed in the previous section.

Let ๐‘†โ„Ž denote a uniform partition of ฮฉ and let ๐‘‰โ„Ž denote the space of continuous piecewise linear functions defined on ๐‘†โ„Ž. In order to implement the Galerkin finite element approximation, we adapt finite element discretization on the spatial axis and the backward Euler finite difference scheme along the temporal axis. We associate shape functions of space ๐‘‰โ„Ž with the standard basis of the functions on the uniform interval with length โ„Ž.

Example 5.1. We first consider a space fractional linear diffusion equation: ๐œ•๐‘ข(๐‘ฅ,๐‘ก)๐œ•๐‘ก=โˆ‡๐œ‡๐‘ข(๐‘ฅ,๐‘ก)+2๐‘ก๐‘ก2๎€ท๐‘ก+1๐‘ข(๐‘ฅ,๐‘ก)โˆ’2๎€ธร—๎ƒฉ๎€ฝ๐‘ฅ+12โˆ’๐œ‡+(1โˆ’๐‘ฅ)2โˆ’๐œ‡๎€พฮ“โˆ’6๎€ฝ๐‘ฅ(3โˆ’๐œ‡)3โˆ’๐œ‡+(1โˆ’๐‘ฅ)3โˆ’๐œ‡๎€พฮ“+๎€ฝ๐‘ฅ(4โˆ’๐œ‡)124โˆ’๐œ‡+(1โˆ’๐‘ฅ)4โˆ’๐œ‡๎€พฮ“๎ƒช(5โˆ’๐œ‡)(5.1) with an initial condition ๐‘ข(๐‘ฅ,0)=๐‘ฅ2(1โˆ’๐‘ฅ)2[],๐‘ฅโˆˆ0,1(5.2) and boundary conditions ๐‘ข(0,๐‘ก)=๐‘ข(1,๐‘ก)=0.(5.3) In this case, the exact solution is ๎€ท๐‘ก๐‘ข(๐‘ฅ,๐‘ก)=2๎€ธ๐‘ฅ+12(1โˆ’๐‘ฅ)2.(5.4)

Tables 1, 2, and 4 show the order of convergence and ๐ฟ2-error between the exact solution and the Galerkin approximate solution of the fully discrete backward Euler method for (5.1) when ๐œ‡=1.6, ๐œ‡=1.8 and ๐œ‡=1.5, respectively. For numerical computation, the temporal step size ๐‘˜=0.001 is used in all three cases. Table 3 shows ๐ฟ2-errors and orders of convergence for the Galerkin approximate solution when ๐œ‡=1.8 and the spatial step size โ„Ž=0.0625.

tab1
Table 1: ๐ฟ2-error and order of convergence in ๐‘ฅ when ๐œ‡=1.6.
tab2
Table 2: ๐ฟ2-error and order of convergence in ๐‘ฅ when ๐œ‡=1.8.
tab3
Table 3: ๐ฟ2-error and order of convergence in ๐‘ก when ๐œ‡=1.8.
tab4
Table 4: ๐ฟ2-error and order of convergence in ๐‘ฅ when ๐œ‡=1.5.

According to Tables 1โ€“3, we may find the order of convergence of ๐‘‚(๐‘˜+โ„Ž๐œ‡) for this linear fractional diffusion problem (5.1)โ€“(5.3) when ๐œ‡โ‰ 3/2. Furthermore, Table 4 shows orders of numerical convergence for the problem when ๐œ‡=3/2, where we may see that the order of convergence is of ๐‘‚(๐‘˜+โ„Ž๐œ‡โˆ’๐œ–), 0<๐œ–<1/2. It follows from Tables 1โ€“4 that numerical computations confirm the theoretical results.

We plot the exact solution and approximate solutions obtained by the backward Euler Galerkin method using โ„Ž=1/32 and ๐‘˜=1/1000 for (5.1) with ๐œ‡=1.6 and ๐œ‡=1.8. Figure 1 shows the contour plots of an exact solution and numerical solutions at ๐‘ก=1, and Figure 2 shows log-log graph for the order of convergence.

596184.fig.001
Figure 1: Exact and numerical solutions with ๐œ‡=1.6 and ๐œ‡=1.8.
596184.fig.002
Figure 2: Log-log plots of the error for the rate of convergence.

Example 5.2. We consider a space fractional diffusion equation with a nonlinear Fisher type source term which is described as ๐œ•๐‘ข(๐‘ฅ,๐‘ก)๐œ•๐‘ก=๐œ…๐œ‡โˆ‡๐œ‡๐‘ข(๐‘ฅ,๐‘ก)+๐œ†๐‘ข(๐‘ฅ,๐‘ก)(1โˆ’๐›ฝ๐‘ข(๐‘ฅ,๐‘ก))(5.5) with an initial condition ๐‘ข(๐‘ฅ,0)=๐‘ข0(๐‘ฅ)(5.6) and boundary conditions ๐‘ข(โˆ’1,๐‘ก)=๐‘ข(1,๐‘ก)=0.(5.7) In fact, we will consider the case of ๐œ…๐œ‡=0.1, ๐›ฝ=1 in (5.5) with an initial condition ๐‘ข0๎‚ป๐‘’(๐‘ฅ)=โˆ’10๐‘ฅ๐‘’,๐‘ฅโ‰ฅ0,10๐‘ฅ,๐‘ฅ<0.(5.8)

For numerical computations, we have to take care of the nonlinear term ๐‘“(๐‘ข)=๐œ†๐‘ข(1โˆ’๐›ฝ๐‘ข). This gives a complicated nonlinear matrix. In order to avoid the difficulty of solving nonlinear system, we adopted a linearized method replacing ๐œ†๐‘ข๐‘›+1(1โˆ’๐›ฝ๐‘ข๐‘›+1) by ๐œ†๐‘ข๐‘›+1(1โˆ’๐›ฝ๐‘ข๐‘›). Figure 3 shows contour plots of numerical solutions at ๐‘ก=1 for (5.5)โ€“(5.8) with ๐œ†=0.25. For numerical computations, step sizes โ„Ž=0.01 and ๐‘˜=0.005 are used. From the numerical results we may find that numerical solutions converge to the solution of classical diffusion equation as ๐œ‡ approaches to 2.

596184.fig.003
Figure 3: Numerical solutions for (5.5) with (5.8).

Example 5.3. We now consider (5.5) with ๐œ…๐œ‡=0.1, ๐›ฝ=1 and boundary conditions lim|๐‘ฅ|โ†’โˆž๐‘ข(๐‘ฅ,๐‘ก)=0.(5.9) We will consider an initial condition with a sharp peak in the middle as ๐‘ข0(๐‘ฅ)=sech2(10๐‘ฅ)(5.10) and an initial condition with a flat roof in the middle as ๐‘ข0โŽงโŽชโŽจโŽชโŽฉ๐‘’(๐‘ฅ)=โˆ’10(๐‘ฅโˆ’1)๐‘’,๐‘ฅ>1,1,โˆ’1<๐‘ฅโ‰ค1,10(๐‘ฅ+1),๐‘ฅโ‰คโˆ’1.(5.11)

Tang and Weber [30] have obtained computational solutions for (5.5) with initial conditions (5.10) and (5.11) using a Petrov-Galerkin method when (5.5) is a classical diffusion problem. We obtain computational results using the method as in Example 5.2. Figure 4 shows contour plots of numerical solutions at ๐‘ก=1 for (5.5) with an initial condition (5.10) when ฮฉ=(โˆ’2,2) and ๐œ†=0.25. Figure 5 shows also contour plots of numerical solutions at ๐‘ก=4 for (5.5) and (5.10) when ฮฉ=(โˆ’4,4) and ๐œ†=1. In both cases, step sizes โ„Ž=0.01 and ๐‘˜=0.005 are used for computation. According to Figures 4 and 5, we may see that the diffusivity depends on ๐œ‡ but it is far less than that of the classical solution. That is, the fractional diffusion problem keeps the peak in the middle for longer time than the classical one does.

596184.fig.004
Figure 4: Numerical solutions at ๐‘ก=1 for (5.5) and (5.10) with ๐œ†=0.25.
596184.fig.005
Figure 5: Numerical solutions at ๐‘ก=4 for (5.5) and (5.10) with ๐œ†=1.

Figure 6 shows contour plots of numerical solutions for (5.5) with an initial condition (5.10) when ๐œ‡=1.8, ฮฉ=(โˆ’2,2) and ๐œ†=1. In this case, step sizes โ„Ž=0.01 and ๐‘˜=0.005 are also used for computation. But the period of time is from ๐‘ก=0 to ๐‘ก=5. According to Figure 6, we may see that the peak goes down rapidly for a short time, and it begins to go up after the contour arrives at the lowest level.

596184.fig.006
Figure 6: Numerical solutions for (5.5) and (5.10) with ๐œ†=1.

Figure 7 shows contour plots of numerical solutions at ๐‘ก=1 for (5.5) with an initial condition (5.11) when ฮฉ=(โˆ’4,4) and ๐œ†=0.25. In this case, step sizes โ„Ž=0.01 and ๐‘˜=0.005 are also used for computation. According to Figure 7, we may find that the fractional diffusion problem keeps the flat roof in the middle for longer time than the classical one does.

596184.fig.007
Figure 7: Numerical solutions for (5.5) and (5.11) with ๐œ†=0.25.

6. Concluding Remarks

Galerkin finite element methods are considered for the space fractional diffusion equation with a nonlinear source term. We have derived the variational formula of the semidiscrete scheme by using the Galerkin finite element method in space. We showed existence and stability of solutions for the semidiscrete scheme. Furthermore, we derived the fully time-space discrete variational formulation using the backward Euler method. Existence and uniqueness of solutions for the fully discrete Galerkin method have been discussed. Also we proved that the scheme is unconditionally stable, and it has the order of convergence of )๐‘‚(๐‘˜+โ„Žฬƒ๐›พ, where ฬƒ๐›พ is a constant depending on the order of fractional derivative. Numerical computations confirm the theoretical results discussed in the previous section for the problem with a linear source term. For the fractional diffusion problem with a nonlinear source term, we may find that the diffusivity depends on the order of fractional derivative, and numerical solutions of fractional order problems are less diffusive than the solution of a classical diffusion problem.

Acknowledgment

The authors would like to express sincere thanks to the referee for their invaluable comments.

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