Analysis Of Neutral Delay Parabolic Equation By Virtual Element Method

Delayed parabolic equation is a kind of differential equation with time delay term,which is widely used in statistics,finance,physical chemistry,biology,aerospace and other fields.And neutral delay differential equations is delay differential equation of a kind of important equation,the equation and the current state and not only in the past the status of the relevant period of time,and also related to the rate of change of the past,it brings to the solution of the equation is difficult,so the neutral delay parabolic equation numerical solution method is meaningful.The virtual element method can be regarded as the extension of the finite element method and has some advantages,such as flexible mesh processing and avoiding explicit construction of shape function.In this paper,the virtual element method is used to solve a class of neutral delay parabolic equations with time delay term.The coordinated virtual element method and the noncoordinated virtual element method are used respectively.The error analysis results in the semi-discrete and fully discrete schemes are given.In the first part,we discuss the coordination virtual element analysis of the neutral delay parabolic equation.Firstly,we give the shape function space,select the appropriate degree of freedom,define the bilinear form,and satisfy the consistency and stability properties.Then two lemmas are given,and the results of error estimation without delay term and with time delay term are obtained.The semi-discrete format is given,and the optimal results of error estimation are obtained.Finally,using backward Euler method,the fully discrete format is obtained,and the error estimation results are given.In the second part,we discuss the nonconsonant virtual element analysis of neutral delay parabolic equations.First,we give the nonconsonant virtual element function space,degree of freedom,and define the virtual element space and bilinear form.Then,on the basis of analyzing the incongruity tolerance error,we prove that the conclusions of the two lemmas are still valid in the case of incongruity.Finally,the error estimation results of the nonconcordant virtual element method in both semi-discrete and fully discrete schemes are given.