Postprocessing Of Galerkin Methods For Delay Differential Equations Of Pantograph Type

As one of the important mathematical models,the delay differential equations(DDEs)of pantograph type have been widely used in many fields such as engineering,biology systems,physics and medicine.Scholars of these fields hope to use this mathematical model to solve real practical problems.Due to the difficulty to obtain the exact solutions of these equations,it is meaningful to study the numerical solutions both in theory and in application.The finite element method is a kind of general and efficient method in solving these problems.It is a perfect combination of classical variational methods and piecewise polynomial interpolation theory,which have a high accuracy,good numerical stability and convergence.Researching the convergence of this kind of equations is of great significance and practice.We can use Galerkin methods to solve DDEs of pantograph type,and obtain the optimal global convergence results and the local superconvergence with quasi-geometric meshes.In this thesis,we propose several methods to process the Galerkin solutions for delay differential equations of pantograph type.The innovative of this article is that the postprocessing method improved the global convergence order of solutions and proved the approximate solution converging to exact solution of the equations.Firstly,the domestic and overseas development are state and numeral methods for DDEs of pantograph type are introduced.We also discuss the convergence properties of continuous Galerkin(CG)solutions and discontinuous Galerkin(DG)solutions for DDEs with proportional delay.Then we propose several techniques to process the CG solutions for DDEs of pantograph type,which include interpolation postprocessing(integration type,Lagrange type,polynomial preserving recovery type)and iteration postprocessing.By using this postprocessing methods,the global convergence order of solutions will have an improvement.In addition,we investigate several techniques to process the DG solutions for DDEs of pantograph type,which also include interpolation postprocessing and iteration postprocessing.By using these postprocessing methods,the global convergence order of solutions will have an improvement.At last,based on the nodal superconvergence results,a higher interpolation operator can be defined by using the interpolation postprocessing technique to process the DG and CG solutions respectively.The global superconvergence order obtained by this postprocessing method is higher than the general iteration interpolation postprocessing method.