时间分数阶扩散方程线性三角形元的高精度分析

摘要/Abstract

摘要：

该文基于线性三角形元和改进的$L1$格式,对具有$\alpha$阶Caputo导数的时间分数阶扩散方程建立了一个全离散逼近格式.首先,证明了该格式的无条件稳定性.其次,利用该单元及Ritz投影算子的性质,导出了关于投影算子具有$O(h^2+\tau^{2-\alpha})$阶的超逼近性质.再结合插值算子和投影算子的关系,进一步导出了关于插值算子具有$O(h^2+\tau^{2-\alpha})$阶的超逼近性质.然后,借助插值后处理技术得到了整体超收敛估计.最后,利用数值算例验证了理论分析的正确性.

关键词: 时间分数阶扩散方程, 线性三角形元, 全离散格式, 无条件稳定, 超逼近和超收敛

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.

Key words: Time fractional diffusion equations, Linear triangular element, Fully-discrete scheme, Unconditional stability, Superclose and superconvergence

中图分类号:

O175.8

史艳华,张亚东,王芬玲,赵艳敏,王萍莉. 时间分数阶扩散方程线性三角形元的高精度分析[J]. 数学物理学报, 2019, 39(4): 839-850.

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

图/表 5

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参考文献 29

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$ 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