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June  2020, 28(2): 897-910. doi: 10.3934/era.2020047

## Superclose analysis of a two-grid finite element scheme for semilinear parabolic integro-differential equations

 School of Mathematics and Statistics, Beihua University, Jilin 132013, Jilin, China

* Corresponding author

Received  February 2020 Revised  April 2020 Published  June 2020

Fund Project: This work is supported by Technology Research Project of Jilin Provincial Department of Education(JJKH20190634KJ)

In this paper, a two-grid finite element scheme for semilinear parabolic integro-differential equations is proposed. In the two-grid scheme, continuous linear element is used for spatial discretization, while Crank-Nicolson scheme and Leap-Frog scheme are ultilized for temporal discretization. Based on the combination of the interpolation and Ritz projection technique, some superclose estimates between the interpolation and the numerical solution in the $H^1$-norm are derived. Notice that we only need to solve nonlinear problem once in the two-grid scheme, namely, the first time step on the coarse-grid space. A numerical example is presented to verify the effectiveness of the proposed two-grid scheme.

Citation: Changling Xu, Tianliang Hou. Superclose analysis of a two-grid finite element scheme for semilinear parabolic integro-differential equations. Electronic Research Archive, 2020, 28 (2) : 897-910. doi: 10.3934/era.2020047
##### References:
 [1] J. R. Cannon and Y. P. Lin, A priori $L^2$ error estimates for finite-element methods for nonlinear diffusion equations with memory, SIAM J. Numer. Anal., 27 (1990), 595-607.  doi: 10.1137/0727036.  Google Scholar [2] L. P. Chen and Y. P. Chen, Two-grid method for nonlinear reaction-diffusion equations by mixed finite element methods, J. Sci. Comput., 49 (2011), 383-401.  doi: 10.1007/s10915-011-9469-3.  Google Scholar [3] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems Ⅳ: Nonlinear problems, SIAM J. Numer. Anal., 32 (1995), 1729-1749.  doi: 10.1137/0732078.  Google Scholar [4] S. M. F. Garcia, Improved error estimates for mixed finite element approximations for nonlinear parabolic equations: The discrete-time case, Numer. Methods Partial Differ. Equ., 10 (1994), 149-169.  doi: 10.1002/num.1690100203.  Google Scholar [5] T. L. Hou, W. Z. Jiang, Y. T. Yang and H. T. Leng, Two-grid $P^2_0$-$P_1$ mixed finite element methods combined with Crank-Nicolson scheme for a class of nonlinear parabolic equations, Appl. Numer. Math., 137 (2019), 136-150.  doi: 10.1016/j.apnum.2018.11.009.  Google Scholar [6] M.-N. Le Roux and V. Thomée, Numerical solution of semilinear integro-differential equations of parabolic type with nonsmooth data, SIAM J. Numer. Anal., 26 (1989), 1291-1309.  doi: 10.1137/0726075.  Google Scholar [7] Q. Lin and J. Lin, Finite Element Methods: Accuracy and Improvement, Science Press, Beijing, 2006.   Google Scholar [8] W. Liu, H. Rui and Y. P. Bao, Two kinds of two-grid algorithms for finite difference solutions of semilinear parabolic equations, J. Sys. Sci. Math. Sci., 30 (2010), 181-190.   Google Scholar [9] P. K. Moore, A posterior error estimation with finite element semi- and fully discrete methods for nonlinear parabolic equations in one space dimension, SIAM J. Numer. Anal., 31 (1994), 149-169.  doi: 10.1137/0731008.  Google Scholar [10] D. Y. Shi and P. C. Mu, Superconvergence analysis of a two-grid method for semilinear parabolic equations, Appl. Math. Lett., 84 (2018), 34-41.  doi: 10.1016/j.aml.2018.04.012.  Google Scholar [11] J. C. Xu, A novel two-grid method for semilinear elliptic equations, SIAM J. Sci. Comput., 15 (1994), 231-237.  doi: 10.1137/0915016.  Google Scholar [12] J. C. Xu, Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal., 33 (1996), 1759-1777.  doi: 10.1137/S0036142992232949.  Google Scholar [13] J. M. Yang and X. Q. Xing, A two-grid discontinuous Galerkin method for a kind of nonlinear parabolic problems, Appl. Math. Comput., 346 (2019), 96-108.  doi: 10.1016/j.amc.2018.09.067.  Google Scholar

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##### References:
 [1] J. R. Cannon and Y. P. Lin, A priori $L^2$ error estimates for finite-element methods for nonlinear diffusion equations with memory, SIAM J. Numer. Anal., 27 (1990), 595-607.  doi: 10.1137/0727036.  Google Scholar [2] L. P. Chen and Y. P. Chen, Two-grid method for nonlinear reaction-diffusion equations by mixed finite element methods, J. Sci. Comput., 49 (2011), 383-401.  doi: 10.1007/s10915-011-9469-3.  Google Scholar [3] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems Ⅳ: Nonlinear problems, SIAM J. Numer. Anal., 32 (1995), 1729-1749.  doi: 10.1137/0732078.  Google Scholar [4] S. M. F. Garcia, Improved error estimates for mixed finite element approximations for nonlinear parabolic equations: The discrete-time case, Numer. Methods Partial Differ. Equ., 10 (1994), 149-169.  doi: 10.1002/num.1690100203.  Google Scholar [5] T. L. Hou, W. Z. Jiang, Y. T. Yang and H. T. Leng, Two-grid $P^2_0$-$P_1$ mixed finite element methods combined with Crank-Nicolson scheme for a class of nonlinear parabolic equations, Appl. Numer. Math., 137 (2019), 136-150.  doi: 10.1016/j.apnum.2018.11.009.  Google Scholar [6] M.-N. Le Roux and V. Thomée, Numerical solution of semilinear integro-differential equations of parabolic type with nonsmooth data, SIAM J. Numer. Anal., 26 (1989), 1291-1309.  doi: 10.1137/0726075.  Google Scholar [7] Q. Lin and J. Lin, Finite Element Methods: Accuracy and Improvement, Science Press, Beijing, 2006.   Google Scholar [8] W. Liu, H. Rui and Y. P. Bao, Two kinds of two-grid algorithms for finite difference solutions of semilinear parabolic equations, J. Sys. Sci. Math. Sci., 30 (2010), 181-190.   Google Scholar [9] P. K. Moore, A posterior error estimation with finite element semi- and fully discrete methods for nonlinear parabolic equations in one space dimension, SIAM J. Numer. Anal., 31 (1994), 149-169.  doi: 10.1137/0731008.  Google Scholar [10] D. Y. Shi and P. C. Mu, Superconvergence analysis of a two-grid method for semilinear parabolic equations, Appl. Math. Lett., 84 (2018), 34-41.  doi: 10.1016/j.aml.2018.04.012.  Google Scholar [11] J. C. Xu, A novel two-grid method for semilinear elliptic equations, SIAM J. Sci. Comput., 15 (1994), 231-237.  doi: 10.1137/0915016.  Google Scholar [12] J. C. Xu, Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal., 33 (1996), 1759-1777.  doi: 10.1137/S0036142992232949.  Google Scholar [13] J. M. Yang and X. Q. Xing, A two-grid discontinuous Galerkin method for a kind of nonlinear parabolic problems, Appl. Math. Comput., 346 (2019), 96-108.  doi: 10.1016/j.amc.2018.09.067.  Google Scholar
The error and the convergence order of $\|u_h^n-I_h u^n\|_1$ at $t = 0.125$ with $h = \Delta t$
 $h$ $\|u_h^n-I_h u^n\|_1$ order $1/32$ 9.6461e-04 - $1/64$ 2.4062e-04 2.00 $1/128$ 6.0130e-05 2.00 $1/256$ 1.5037e-05 2.00
 $h$ $\|u_h^n-I_h u^n\|_1$ order $1/32$ 9.6461e-04 - $1/64$ 2.4062e-04 2.00 $1/128$ 6.0130e-05 2.00 $1/256$ 1.5037e-05 2.00
The error and the convergence order of $\|u_H^n-I_H u^n\|_1$ at $t = 0.0625$ with $H = \Delta t$
 $H$ $\|u_H^n-I_H u^n\|_1$ order $1/16$ 1.9302e-02 - $1/32$ 4.8349e-03 2.00 $1/64$ 1.2096e-03 2.00 $1/128$ 3.0244e-04 2.00
 $H$ $\|u_H^n-I_H u^n\|_1$ order $1/16$ 1.9302e-02 - $1/32$ 4.8349e-03 2.00 $1/64$ 1.2096e-03 2.00 $1/128$ 3.0244e-04 2.00
The error and the convergence order of $\|\widetilde{u}_h^n-I_h u^n\|_1$ at $t = 0.001$ with $\Delta t = 0.0001$ and $h = H^2$
 $H$ $\|\widetilde{u}_h^n-I_h u^n\|_1$ order $1/2$ 8.7973e-04 - $1/4$ 7.1420e-05 3.51 $1/8$ 4.6798e-06 3.91 $1/16$ 2.9328e-07 3.99
 $H$ $\|\widetilde{u}_h^n-I_h u^n\|_1$ order $1/2$ 8.7973e-04 - $1/4$ 7.1420e-05 3.51 $1/8$ 4.6798e-06 3.91 $1/16$ 2.9328e-07 3.99
The cpu time of two-grid scheme and Crank-Nicolson scheme for each time step ($h = \Delta t$)
 $(H,h)$ two-grid time (s) Crank-Nicolson time (s) $(1/4,1/16)$ 0.0998 0.1164 $(1/8,1/64)$ 0.9118 1.2019 $(1/16,1/256)$ 13.6126 17.9624
 $(H,h)$ two-grid time (s) Crank-Nicolson time (s) $(1/4,1/16)$ 0.0998 0.1164 $(1/8,1/64)$ 0.9118 1.2019 $(1/16,1/256)$ 13.6126 17.9624
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