P-version DG Method For Solving Singularly Perturbed Integro-differential Equations

The object of this paper is the singularly perturbed Volterra integro-differential equations(VIDES), which comes from many physical and biological problems, such as diffusion of dissipative process, epidemic dynamics and so on. For this kind of problem, due to the existence of the small parameter ε, the solution undergoes a rapid transition in small narrow region, it is called boundary layer or interior layer phenomenon. On the other hand, the existence of the integral item yields the memorial property of the problem. The study of the high-accuracy method for the singularly perturbed VIDES has to face some challenge both from "Layer" phenomenon and long time behavior. The known results showed that the convergence rate of p-version FEM is twice more than the h-version FEM. It is known that the discontinuous Galerkin(DG) methods adopts completely discontinuous piecewise polynomial space and test function to solve the numerical solution, consequently the choice of degree of freedom is more flexible, its schemes are more locally compact, More importantly, it can simulate the rapid changes of the solution more efficiently.In this paper we mainly focus on the p-version DG method(p-version DG) for the dealing with singularly perturbed Volterra integro-differential equation-s and analyze its convergence property. Firstly, we introduce the numerical scheme of the p-version DG method for solving it, Then the uniform conver-gence property of this approach in the sense of L2 norm is verified theoretically. Furthermore, the numerical results validate our theoretical prediction.