@Article{JCM-40-5, author = {Tianliang, Hou and Liu, Chunmei and Dai, Chunlei and Luoping, Chen and Yin, Yang}, 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 = {https://global-sci.com/article/84197/two-grid-algorithm-of-h1-galerkin-mixed-finite-element-methods-for-semilinear-parabolic-integro-differential-equations} }