The Research On The Finite Element Methods For Delay Differential Equations And On Superconvergence

Delay differential equations play important roles in describing the physical,biological,chemical,control and other problems of many real-world processes.Be-cause the exact solution is not available in general,the numerical simulation is an important part in the area of computational mathematics.This paper investigates the convergence of Galerkin methods for delay differential equations which contain delay differential equations with the variable dependent delay,delay differential equations with state dependent delay,partial differential equations with constant delay.In Chapter 1,we first introduce the study background of the delay differential equations and summarize the research progress of numerical methods for delay differential equations.We then introduce some basic knowledge and main research contents of this paper.Chapter 2 focuses on the continuous Galerkin?CG?method for nonlinear de-lay differential equations?DDEs?with proportional delay.The past result is based on uniform meshes,in which the classical superconvergence cannot be achieved by CG method.This is because the mapping qt of a subinterval(t_n-1,t_n)in general lies in two adjacent previous subintervals(t_i-1,t_i)?[t_i,t_i+1)?i<n-2?and re-sults in the lower convergence rate.In order to avoid this problem,We discuss the convergence of the CG solutions for nonlinear DDEs with proportional delay on quasi-geometric meshes.By an element orthogonal analysis and the lower degree interpolation construction,we get the global convergence and classical local super-convergence results of CG solutions for DDEs with proportional delay.Moreover,it shows that the discretized CG method and the collocation method are identical for linear delay differential equations with proportional delay.Numerical examples are provided to illustrate the theoretical results.In Chapter 3,we propose discontinuous Galerkin?DG?method for nonlinear DDEs with vanishing delay.On quasi-graded meshes,we present the global con-vergence of DG solutions and show that the optimal order of the DG solutions at the mesh points is O(h^2m+1)by constructing an auxiliary problem.By analyzing the supercloseness between the DG solutions and the interpolation?_hu of the exact solution u,we get the optimal order O(h^m+2)of the DG solution at Radau II points.The theoretical results are illustrated by a broad range of numerical examples.In Chapter 4,the DG method is applied to DDEs with non-vanishing state-dependent delay.We extend the convergence results of DG solutions for variable dependent delay differential equations to state dependent delay differential equa-tions.Because the delay term is a function of the exact solution,the relevant numerical solution is very difficult to obtain,and the suitable partition cannot be provided such that classical superconvergence is obtained at mesh points before numerically solving this kind DDEs.We first propose the DG scheme of state dependent DDEs and then propose algorithms to get discontinuous points and the suitable partition.The classical superconvergence of DG solutions is then obtained.Numerical examples are provided to illustrate the theoretical results.In chapter 5,time stepping DG method combined with standard finite el-ement method in space is proposed to solve a class of partial differential equa-tions?PDEs?with time constant delay.The time semi-discretization is derived by the DG method.Based on some Lemmas,we prove global convergence of the time semi-discrete scheme under uniform meshes.We then use the standard Galerkin method in space to obtain the complete discrete scheme and present the optimal global convergence of the full discretization.Numerical experiments for one-dimensional and two-dimensional equations are provided to demonstrate the optimal global convergence.Moreover,we find numerical solutions have lo-cal nodal superconvergence in time,which puts a very good foundation for next research work.The study of chapter 6 is based on Zhang[113].We focus on the approxi-mation properties of polynomial interpolants of analytic functions,and provide approximation of derivative superconvergence points of interpolation polynomial-s.In this chapter,our results provide first derivatives superconvergent points of interpolation polynomials with 14 digits of accuracy which makes it easy to use.Numerical examples are provided to demonstrate the location of superconvergence points of higher precision,especially near the endpoints.