## High Accuracy Analysis of Linear Triangular Element for Time Fractional Diffusion Equations

Shi Yanhua,, Zhang Yadong, Wang Fenling,, Zhao Yanmin, Wang Pingli

 基金资助: 河南省高等学校重点科研项目.  17A110011

 Fund supported: the Key Scientific Research Projects in Universities of Henan Province.  17A110011

Abstract

In this paper, based on linear triangular element and improved $L1$ approximation, a fully-discrete scheme is proposed for time fractional diffusion equations with $\alpha$ order Caputo fractional derivative. Firstly, the unconditional stability is proved. Secondly, by employing the properties of the element and Ritz projection operator, superclose analysis for the projection operator is deduced with order $O(h^2+\tau^{2-\alpha})$. Further more, combining with relationship between the interpolation operator and Ritz projection, superclose analysis for the interpolation operator is also investigated with order $O(h^2+\tau^{2-\alpha})$. And then, the superconvergence result is obtained through the interpolated postprocessing technique. Finally, numerical results are provided to show the validity of our theoretical analysis.

Keywords： Time fractional diffusion equations ; Linear triangular element ; Fully-discrete scheme ; Unconditional stability ; Superclose and superconvergence

Shi Yanhua, Zhang Yadong, Wang Fenling, Zhao Yanmin, Wang Pingli. High Accuracy Analysis of Linear Triangular Element for Time Fractional Diffusion Equations. Acta Mathematica Scientia[J], 2019, 39(4): 839-850 doi:

## 1 引言

$$$\left\{\begin{array}{ll} \displaystyle \frac{\partial^\alpha u }{\partial t^\alpha}-\Delta u=f({\bf x}, t), & ({\bf x}, t)\in\Omega\times(0, T], \\[2mm] u({\bf x}, t)=0, &({\bf x}, t)\in \partial\Omega\times(0, T], \\ { u({\bf x}, 0)}=u_0({\bf x}), & {\bf x}\in\Omega , \end{array}\right.$$$

## 2 全离散逼近格式和一些引理

$$$\left\{\begin{array}{ll}\displaystyle(\frac{\partial^\alpha u}{\partial t^\alpha}, v)+(\nabla u, \nabla v)=(f, v), \quad \forall v\in H_0^1(\Omega), \\[2mm]u(X, 0)=u_0(X).\end{array}\right.$$$

$0={t_0}<{t_1}<\cdots <{t_{N-1}}<{t_N}=T$$[0, T]上步长为\tau=T/N的剖分, t_n=n\tau,$$n=0, 1, 2, \cdots , N$.我们定义

$\begin{eqnarray}\|R^n\|_0\leq C\tau^{2-\alpha}\displaystyle\max\limits_{0\leq t\leq T}\|u_{tt}({\bf x}, t)\|_{0}.\end{eqnarray}$

$$$\left\{\begin{array}{ll}(D_t^\alpha U^n, v_h)+(\nabla U^n, \nabla v_h)=( f^n, v_h), \quad \forall v_h\in V_0^h, \\ \ U^0=R_hu_0({\bf x}), \end{array}\right.$$$

$\begin{eqnarray}(\nabla{\xi}^n, \nabla D_t^\alpha{\xi}^n)&=&\frac{\tau^{-\alpha}}{\Gamma(2-\alpha)}(\nabla{\xi}^n, \sum\limits_{k=0}^nb_k^n\nabla{\xi}^k)\nonumber\\ &=&\displaystyle\frac{\tau^{-\alpha}}{2\Gamma(2-\alpha)}(\|\nabla{\xi}^n\|_0^2+ \displaystyle\sum\limits_{k=0}^{n-1}b_k^n\|\nabla{\xi}^k\|_0^2-\displaystyle\sum\limits_{k=0}^{n-1}b_k^n\|\nabla{\xi}^k-\nabla{\xi}^n\|_0^2). \end{eqnarray}$

$$$|(D_t^\alpha {\eta}^n, D_t^\alpha{\xi}^n)|\leq\displaystyle\frac{1}{2}\|D_t^\alpha{\eta}^n\|_0^2+\displaystyle\frac{1}{2}\|D_t^\alpha{\xi}^n\|_0^2 \leqCh^4\|u_t\|_{L^{\infty}(H^2(\Omega))}^2+\displaystyle\frac{1}{2}\|D_t^\alpha{\xi}^n\|_0^2.$$$

$$$\displaystyle|-(R^n, D_t^\alpha{\xi}^n)|\leq\|R^n\|_0\|D_t^\alpha{\xi}^n\|_0\leqC\tau^{4-2\alpha}\max\limits_{0\leq t\leq T}\|u_{tt}({\bf x}, t)\|_{0}^2+\displaystyle\frac{1}{2}\|D_t^\alpha{\xi}^n\|_0^2.$$$

$\begin{eqnarray}\|\nabla{\xi}^n\|_0^2\leq Ch^4\|u_t\|_{L^{\infty}(H^2(\Omega))}^2+C\tau^{4-2\alpha}\max\limits_{0\leq t\leq T}\|u_{tt}({\bf x}, t)\|_{0}^2.\end{eqnarray}$

由于$(\nabla (R_hu^n-u^n), \nabla(U^n-I_hu^n))=0$,则

 $t_n$ $\tau$ $\|u^n-U^n\|_{0}$ 阶 $\|I_hu^n-U^n\|_{1}$ 阶 $\|u^n-\Pi_{2h}U^n\|_{1}$ 阶 ${t_n}/{2}$ 3.869e-03 / 4.026e-03 / 2.121e-02 / ${t_n}/{4}$ 1.687e-03 1.198 1.780e-03 1.178 9.538e-03 1.153 0.1 ${t_n}/{8}$ 7.486e-04 1.172 7.959e-04 1.161 4.292e-03 1.152 ${t_n}/{16}$ 3.368e-04 1.152 3.570e-04 1.157 1.934e-03 1.150 ${t_n}/{2}$ 2.091e-05 / 1.294e-05 / 1.656e-04 / ${t_n}/{4}$ 9.240e-06 1.178 5.780e-06 1.163 7.369e-05 1.168 0.01 ${t_n}/{8}$ 4.125e-06 1.164 2.596e-06 1.155 3.308e-05 1.156 ${t_n}/{16}$ 1.855e-06 1.153 1.169e-06 1.151 1.490e-05 1.150

 $t_n$ $\tau$ $\|u^n-U^n\|_{0}$ 阶 $\|I_hu^n-U^n\|_{1}$ 阶 $\|u^n-\Pi_{2h}U^n\|_{1}$ 阶 ${t_n}/{2}$ 6.913e-04 / 9.330e-04 / 2.740e-03 / ${t_n}/{4}$ 1.890e-04 1.871 2.537e-04 1.879 7.466e-04 1.876 0.1 ${t_n}/{8}$ 5.174e-05 1.869 6.974e-05 1.863 2.044e-04 1.869 ${t_n}/{16}$ 1.430e-05 1.856 1.9267e-05 1.856 5.649e-05 1.855 ${t_n}/{2}$ 6.845e-05 / 8.960e-05 / 2.832e-04 / ${t_n}/{4}$ 1.881e-05 1.864 2.461e-05 1.864 7.790e-05 1.862 0.01 ${t_n}/{8}$ 5.196e-06 1.856 6.799e-06 1.856 2.153e-05 1.855 ${t_n}/{16}$ 1.440e-06 1.851 1.885e-06 1.851 5.968e-06 1.851

 $\alpha$ $h$ $t_{n}=0.2$ 阶 $t_{n}=0.4$ 阶 $t_{n}=0.7$ 阶 $t_{n}=0.9$ 阶 1/4 3.952e-02 / 8.084e-02 / 1.437e-01 / 1.860e-01 / $1/8$ 1.042e-02 1.923 2.140e-02 1.918 3.814e-02 1.914 4.942e-02 1.912 0.3 $1/16$ 2.640e-03 1.981 5.424e-03 1.980 9.676e-03 1.979 1.254e-02 1.979 $1/32$ 6.622e-04 1.995 1.361e-03 1.995 2.428e-03 1.995 3.146e-03 1.995 1/4 3.642e-02 / 7.756e-02 / 1.413e-01 / 1.844e-01 / 1/8 9.487e-02 1.941 2.039e-02 1.928 3.737e-02 1.919 4.889e-02 1.915 0.6 $1/16$ 2.395e-03 1.986 5.158e-03 1.983 9.471e-03 1.980 1.240e-02 1.979 $1/32$ 6.002e-04 1.997 1.293e-03 1.995 2.376e-03 1.995 3.111e-03 1.995 $1/4$ 3.229e-02 / 7.331e-02 / 1.388e-01 / 1.833e-01 / $1/8$ 8.277e-03 1.964 1.908e-02 1.942 3.656e-02 1.925 4.851e-02 1.918 0.9 $1/16$ 2.081e-03 1.992 4.814e-03 1.986 9.255e-03 1.982 1.230e-02 1.980 $1/32$ 5.209e-04 1.998 1.206e-03 1.997 2.321e-03 1.996 3.084e-03 1.995

 $\alpha$ $h$ $t_{n}=0.2$ 阶 $t_{n}=0.4$ 阶 $t_{n}=0.7$ 阶 $t_{n}=0.9$ 阶 $1/4$ 5.310e-02 / 1.103e-01 / 1.982e-01 / 2.575e-01 / $1/8$ 1.387e-02 1.937 2.894e-02 1.930 5.217e-02 1.925 6.790e-02 1.923 0.3 $1/16$ 3.506e-03 1.984 7.324e-03 1.982 1.321e-02 1.981 1.720e-02 1.981 $1/32$ 8.791e-04 1.995 1.837e-03 1.996 3.315e-03 1.995 4.316e-03 1.995 $1/4$ 4.606e-02 / 1.028e-01 / 1.926e-01 / 2.538e-01 / $1/8$ 1.183e-02 1.961 2.673e-02 1.944 5.048e-02 1.932 6.676e-02 1.927 0.6 $1/16$ 2.977e-03 1.990 6.749e-03 1.986 1.277e-02 1.983 1.690e-02 1.982 $1/32$ 7.457e-04 1.997 1.691e-03 1.996 3.202e-03 1.996 4.240e-03 1.995 $1/4$ 3.674e-02 / 9.320e-02 / 1.870e-01 / 2.514e-01 / $1/8$ 9.223e-03 1.994 2.389e-02 1.964 4.872e-02 1.940 6.593e-02 1.931 0.9 $1/16$ 2.311e-03 1.997 6.012e-03 1.991 1.231e-02 1.985 1.668e-02 1.983 $1/32$ 5.781e-04 1.999 1.505e-03 1.998 3.085e-03 1.996 4.182e-03 1.996

 $\alpha$ $h$ $t_{n}=0.2$ 阶 $t_{n}=0.4$ 阶 $t_{n}=0.7$ 阶 $t_{n}=0.9$ 阶 $1/4$ 1.540e-01 / 3.080e-01 / 5.390e-01 / 6.931e-01 / $1/8$ 4.179e-02 1.882 8.357e-02 1.882 1.462e-01 1.882 1.880e-01 1.882 0.3 $1/16$ 1.067e-02 1.970 2.133e-02 1.970 3.733e-02 1.970 4.800e-02 1.970 $1/32$ 2.681e-03 1.992 5.361e-03 1.992 9.383e-03 1.992 1.206e-02 1.992 $1/4$ 1.545e-01 / 3.082e-01 / 5.391e-01 / 6.931e-01 / $1/8$ 4.190e-02 1.882 8.362e-02 1.882 1.462e-01 1.882 1.880e-01 1.882 0.6 $1/16$ 1.069e-02 1.970 2.134e-02 1.970 3.733e-02 1.970 4.780e-02 1.970 $1/32$ 2.687e-03 1.993 5.364e-03 1.992 9.382e-03 1.992 1.206e-02 1.992 $1/4$ 1.557e-01 / 3.088e-01 / 5.391e-01 / 6.931e-01 / $1/8$ 4.221e-02 1.883 8.379e-02 1.882 1.463e-01 1.882 1.880e-01 1.882 0.9 $1/16$ 1.077e-02 1.971 2.138e-02 1.970 3.734e-02 1.970 4.800e-02 1.970 $1/32$ 2.071e-03 1.993 5.374e-03 1.993 9.384e-03 1.992 1.206e-02 1.992

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

He J H. Nonlinear Oscillation with Fractional Derivative and its Applications//Wen B C. International Conference on Vibrating Engineering' 98. Shenyang:Northeastern Univ Press, 1998:288-291

Ming C , Liu F , Zheng L , et al.

Analytical solutions of multi-term time fractional differential equations and application to unsteady flows of generalized viscoelastic fluid

Computational Methods in Applied Mathematics, 2016, 72: 2084- 2097

He J H .

Approximate analytical solution for seepage flow with fractional derivatives in porous media

Computer Methods in Applied Mechanics & Engineering, 1998, 167 (1-2): 57- 68

Mainardi F. Fractional Calculus, Some Basic Problems in Continuumand Statisticalmechanics//Carpinteri A, Mainardi F, et al. Fractals and Fractional Calculus in Continuum Mechanics. New York:Springer Verlag, 1997:291-348

Kilbas A, Srivastava H, Trujillo J. Theory and Applications of Fractional Differential Equations. Amsterdam:Elsevier, 2006

Liu F , Zhuang P , Liu Q . Numerical Methods of Fractional Partial Differential Equations and Applications. Beijing: Science Press, 2015

Zhang Y N , Sun Z Z , Liao H L .

Finite difference methods for the time fractional diffusion equations and non-uniform meshes

Journal of Computational Physics, 2014, 265 (3): 195- 210

Ye H , Liu F , Anh V .

Compact difference scheme for distributed-order time-fractional diffusion-wave equation on bounded domains

Journal of Computational Physics, 2015, 98: 652- 660

Zeng F , Liu F , Li C , et al.

Crank-nicolson ADI spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation

SIAM Journal on Numerical Analysis, 2014, 52 (6): 2599- 2622

Zheng M , Liu F , Anh V , et al.

A high-order spectral method for the multi-term time-fractional diffusion equations

Applied Mathematical Modelling, 2016, 40 (7-8): 4970- 4985

Liu F , Zhuang P , Turner I , et al.

A new fractional finite volume method for solving the fractional diffusion equation

Applied Mathematical Modelling, 2014, 38 (15-16): 3871- 3878

Jia J , Wang H .

A fast finite volume method for conservative space-fractional diffusion equations in convex domains

Journal of Computational Physics, 2016, 310: 63- 84

Wei L .

Analysis of a new finite difference/local discontinuous Galerkin method for the fractional diffusionwave equation

Numerical Algorithms, 2017, 304: 180- 189

Jin B , Lazarov R , Liu Y , et al.

The Galerkin finite element method for a multi-term time-fractional diffusion equation

Journal of Computational Physics, 2015, 281: 825- 843

Zhuang P , Liu F , Turner I , et al.

Galerkin finite element method and error analysis for the fractional cable equation

Numerical Algorithms, 2016, 72 (2): 447- 466

Metzler R , Klafter J .

The randomwalk's guide to anomalous diffusion:a fractional dynamics approach

Physics Reports, 2000, 339 (1): 1- 77

Lin Q , Tobiska L , Zhou A H .

Superconvergence and extrapolation on nonconforming low order finite elements applied to the Poission equation

IMA Journal of Numerical Analysis, 2005, 25 (1): 160- 181

Shi D Y , Shi Y H .

The lowest order H1-Galerkin mixed finite element method for semi-linear pseudohyperbolic equation

Journal of Systems Science and Mathematical Sciences, 2015, 35 (5): 514- 526

Zhang T .

Superconvergence of finite element approximations to integro-differential equations of parabolic type

Numerical Mathematies A Journal of Chinese Universities, 2001, 23 (3): 193- 201

Shi D Y , Wang J J .

Superconvergence analysis of an H1-Galerkin mixed finite element method for Sobolev equations

Computers & Mathematics with Applications, 2016, 72 (6): 1590- 1602

Shi D Y , Yang H J .

A new approach of superconvergence analysis for nonlinear BBM equation on anisotropic meshes

Applied Mathematics Letters, 2016, 58: 74- 80

Zhao Y M , Zhang Y D , Shi D Y , et al.

Superconvergence analysis of nonconforming fnite element method for two-dimensional time fractional difusion equations

Applied Mathematics Letters, 2016, 59: 38- 47

Zhao Y M , Chen P , Bu W P , et al.

Two mixed finite element methods for time-fractional diffusion equations

Journal of Scientific Computing, 2017, 70 (1): 407- 428

Zhao Y M , Zhang Y D , Liu F , et al.

Analytical solution and nonconforming finite element approximation for the 2D multi-term fractional subdiffusion equation

Applied Mathematical Modelling, 2016, 40: 8810- 8825

Shi D Y , Liang H .

The superconvergence analysis of linear triangular element on anisotropic meshes

Chinese Journal of Engineering Mathematics, 2007, 24 (3): 487- 493

Shi D Y , Wang F L , Zhao Y M .

A new pattern of high accuracy analysis of anisotropic linear element for nonlinear sine-gordon equations

Mathematica Numerica Sinica, 2014, 36 (3): 245- 256

Shi D Y , Wang P L , Zhao Y M .

Superconvergence analysis of anisotropic linear triangular finite element for nonlinear Schrödinger equation

Applied Mathematics Letters, 2014, 38 (38): 129- 134

Lin Q , Y N N . The Construction and Analysis of High Eficient Finite Element Methods. Baoding: Hebei University Press, 1996

Zhang T . Finite Element Methods for Partial Differentio-Integral Equations. Beijing: Science Press, 2009

/

 〈 〉