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 $h=H^2$.