时间分数阶扩散方程线性三角形元的高精度分析

Abstract

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

CLC Number:

O175.8

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.

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Figures/Tables 5

References 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

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

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