A Two-grid Finite Element Method For Nonlinear Parabolic Integro-differential Equations

The finite element method is originally called the matrix approximation method.Based on classical Ritz-Galerkin variational method,the finite element method utilizes piecewise inter-polation polynomials and is a strong and powerful means combined with the development and extension of computer for solving differential equations.The finite element method can not only adapt to various complex regional shapes,but also can get a higher calculation precision.In addition,it also is an effective means of engineering analysis and a general calculation program can be worked out.In this paper,we mainly study the semi-discrete and fully discrete two-grid finite element method for two-dimensional nonlinear parabolic integro-differential equation.We construct the two-grid algorithm of finite element method and give the corresponding error estimates and proofs with the properties of Ritz-Volterra projection.And The numerical examples and the analysis of results are given.First of all,two linear finite element spaces(1and(1_?are selected.On the coarse space(1,we obtain the solution of the fully nonlinear system by finite element discretization.Secondly,we treat the obtained solution as the initial approximation of the solution on the fine grid to linearize the problem.Thus solve the linearized problem on the fine space(1_?and carry out the error estimates.Finally,freefem++is used to program and two numerical examples are given to validate the theoretical results.The numerical examples show that the two-grid finite element method can maintain the same accuracy compared with the standard finite element method but with less computational time.The structure of this paper as follow:Firstly,the purpose of this thesis and research status at home and abroad of finite element method and the two-grid finite element method are introduced.Secondly,We describe the preparatory knowledge needed in this paper.Then we recount the properties of the Ritz-Volterra projection and the optimal error estimates in~1-norm of semi-discrete finite element method and the two-grid algorithm for the nonlinear parabolic integro-differential equations.Thirdly,we construct the fully finite element method and the corresponding two-grid finite element method and give the corresponding prior error estimation and proofs.Finally,we give two numerical examples.