In this paper,we present a two-grid discretization scheme for semilinear parabolic integro-differential equations by Hl-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=H2.

Tianliang Hou;Chunmei Liu;Chunlei Dai;Luoping Chen;Yin Yang

School of Mathematics and Statistics,Beihua University,Jilin 132013,China;College of Science,Hunan University of Science and Engineering,Yongzhou 425199,China;School of Mathematics,Southwest Jiaotong University,Chengdu 611756,China;Hunan Key Laboratory for Computation and Simulation in Science and Engineering,Key Laboratory of Intelligent Computing&Information Processing of Ministry of Education,School of Mathematics and Computational Science,Xiangtan University,Xiangtan 411105,China

计算数学（英文版）

[1]Tianliang Hou;Chunmei Liu;Chunlei Dai;Luoping Chen;Yin Yang-.TWO-GRID ALGORITHM OF H1-GALERKIN MIXED FINITE ELEMENT METHODS FOR SEMILINEAR PARABOLIC INTEGRO-DIFFERENTIAL EQUATIONS)[J].计算数学（英文版）,2022(05):667-685

ALGORITHM,GALERKIN,FINITE,ELEMENT,SEMILINEAR,PARABOLIC,INTEGRO,DIFFERENTIAL,EQUATIONS,superclose

TWO,GRID,OF,H1,MIXED,METHODS,FOR,In,this,paper,two,grid,discretization,scheme,semilinear,parabolic,integro,differential,equations,by,Hl,Galerkin,mixed,methods,We,use,lowest,order,Raviart,Thomas,elements,continuous,spatial,backward,Euler,temporal,Firstly,priori,error,estimates,some,properties,are,derived,Secondly,presented,its,convergence,discussed,proposed,solution,nonlinear,system,fine,reduced,much,coarser,symmetric,positive,definite,algebraic,resulting,still,maintains,optimal,accuracy,Finally,numerical,experiment,implemented,verify,theoretical,results,show,that,achieves,same,property,one,choice,H2

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