Volume 40, Issue 5
Two-Grid Algorithm of $H^{1}$-Galerkin Mixed Finite Element Methods for Semilinear Parabolic Integro-Differential Equations

J. Comp. Math., 40 (2022), pp. 667-685.

Published online: 2022-05

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• Abstract

In this paper, we present a two-grid discretization scheme for semilinear parabolic integro-differential equations by $H^{1}$-Galerkin mixed finite element methods. We use the lowest order Raviart-Thomas mixed finite elements and continuous linear finite element for spatial discretization, and backward Euler scheme for temporal discretization. Firstly, a priori error estimates and some superclose properties are derived. Secondly, a two-grid scheme is presented and its convergence is discussed. In the proposed two-grid scheme, the solution of the nonlinear system on a fine grid is reduced to the solution of the nonlinear system on a much coarser grid and the solution of two symmetric and positive definite linear algebraic equations on the fine grid and the resulting solution still maintains optimal accuracy. Finally, a numerical experiment is implemented to verify theoretical results of the proposed scheme. The theoretical and numerical results show that the two-grid method achieves the same convergence property as the one-grid method with the choice $h=H^2$.

• Keywords

Semilinear parabolic integro-differential equations, H$^1$-Galerkin mixed finite element method, A priori error estimates, Two-grid, Superclose.

49J20, 65N30

htlchb@163.com (Tianliang Hou)

liuchunmei0629@163.com (Chunmei Liu)

1573924707@qq.com (Chunlei Dai)

cherrychen@home.swjtu.edu.cn (Luoping Chen)

yangyinxtu@xtu.edu.cn (Yin Yang)

• BibTex
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@Article{JCM-40-667, author = {Hou , TianliangLiu , ChunmeiDai , ChunleiChen , Luoping and Yang , Yin}, title = {Two-Grid Algorithm of $H^{1}$-Galerkin Mixed Finite Element Methods for Semilinear Parabolic Integro-Differential Equations}, journal = {Journal of Computational Mathematics}, year = {2022}, volume = {40}, number = {5}, pages = {667--685}, abstract = {

In this paper, we present a two-grid discretization scheme for semilinear parabolic integro-differential equations by $H^{1}$-Galerkin mixed finite element methods. We use the lowest order Raviart-Thomas mixed finite elements and continuous linear finite element for spatial discretization, and backward Euler scheme for temporal discretization. Firstly, a priori error estimates and some superclose properties are derived. Secondly, a two-grid scheme is presented and its convergence is discussed. In the proposed two-grid scheme, the solution of the nonlinear system on a fine grid is reduced to the solution of the nonlinear system on a much coarser grid and the solution of two symmetric and positive definite linear algebraic equations on the fine grid and the resulting solution still maintains optimal accuracy. Finally, a numerical experiment is implemented to verify theoretical results of the proposed scheme. The theoretical and numerical results show that the two-grid method achieves the same convergence property as the one-grid method with the choice $h=H^2$.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2101-m2019-0159}, url = {http://global-sci.org/intro/article_detail/jcm/20542.html} }
TY - JOUR T1 - Two-Grid Algorithm of $H^{1}$-Galerkin Mixed Finite Element Methods for Semilinear Parabolic Integro-Differential Equations AU - Hou , Tianliang AU - Liu , Chunmei AU - Dai , Chunlei AU - Chen , Luoping AU - Yang , Yin JO - Journal of Computational Mathematics VL - 5 SP - 667 EP - 685 PY - 2022 DA - 2022/05 SN - 40 DO - http://doi.org/10.4208/jcm.2101-m2019-0159 UR - https://global-sci.org/intro/article_detail/jcm/20542.html KW - Semilinear parabolic integro-differential equations, H$^1$-Galerkin mixed finite element method, A priori error estimates, Two-grid, Superclose. AB -

In this paper, we present a two-grid discretization scheme for semilinear parabolic integro-differential equations by $H^{1}$-Galerkin mixed finite element methods. We use the lowest order Raviart-Thomas mixed finite elements and continuous linear finite element for spatial discretization, and backward Euler scheme for temporal discretization. Firstly, a priori error estimates and some superclose properties are derived. Secondly, a two-grid scheme is presented and its convergence is discussed. In the proposed two-grid scheme, the solution of the nonlinear system on a fine grid is reduced to the solution of the nonlinear system on a much coarser grid and the solution of two symmetric and positive definite linear algebraic equations on the fine grid and the resulting solution still maintains optimal accuracy. Finally, a numerical experiment is implemented to verify theoretical results of the proposed scheme. The theoretical and numerical results show that the two-grid method achieves the same convergence property as the one-grid method with the choice $h=H^2$.

Tianliang Hou, Chunmei Liu, Chunlei Dai, Luoping Chen & Yin Yang. (2022). Two-Grid Algorithm of $H^{1}$-Galerkin Mixed Finite Element Methods for Semilinear Parabolic Integro-Differential Equations. Journal of Computational Mathematics. 40 (5). 667-685. doi:10.4208/jcm.2101-m2019-0159
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