Exponential Time Differencing-padé Finite Element Method For Convection-diffusion-reaction Equations With Time Delay

Delay partial differential equations(DPDEs)are widely used in economics,physics,ecology,biological systems,pharmacology,epidemiology,engineering control,computer aided design,nuclear engineering,climate models etc.,and have been getting more and more attention.It is very meaningful to solve the high precision approximate solution of delay partial differential equations in real life.Therefore,investigations on numerical methods are significant for solving this kind of problem.In this article,the research object is the parabolic convection-diffusion-reaction equation with time delay,which is important for DPDEs.In this paper,ETD3-Padéand ETD4-Padéfinite element methods are proposed and ana-lyzed for nonlinear delayed convection-diffusion-reaction equations with Dirichlet boundary conditions.Discretizing the PDE with ETD-Padémethod in time and using finite element method in space.The whole investigation can be separated into five parts.Firstly,starting from the original equation,writing the true so-lution expression of the equation by using Constant transform method,due to the ETD3 and ETD4 numerical calculation schemes contain the exponent oper-ator e^{-k A}and operator A of higher order inverse,which lead to a considerable difficulty in practical calculations,in order to overcome these difficulties,Padéap-proximation is used for exponential operators,corresponding ETD-Padénumerical calculation formats are obtained.Secondly,under the assumption of global Lips-chitz continuity,the unconditional numerical stability of the numerical scheme is proved.Thirdly,the numerical scheme was spatially discretized to obtain the fully discrete numerical calculation scheme,convergence of the fully discrete numerical scheme is analyzed,which gives the convergence order of O(k~3+h^r)(ETD3-Padé)or O(k~4+h^r)(ETD4-Padé)in the L~2norm.Next,Based on ETD-Padénumerical computation scheme,fast algorithms of numerical computation schemes are pro-posed.Finally,the stability and convergence of the numerical calculation scheme are verified by numerical experimental results.At the end of this paper,the follow-up research direction will be put forward according to the new problems in the research process of this topic.