Collocation Methods And Error Estimates For Drift-diffusion Models

The collocation method is a numerical method using piecewise polynomial satisfied at some certain points to approximate solution. Originally we chose natural nodes to solve problems. But its precision is not high enough. To accelerate the convergence speed, we began to choose Gaussian points.The collocation method at gaussian points is first presented by Boor and Swartz. It’s a kind of effective numerical method in recent decades. Since it does not form the stiffness matrix and no need to compute numerical inte-gration, with the advantages of easy calculation, high order of convergence, the method has been widely used in many important areas of engineering and scientific computing.Semiconductor mathematical model is one of the important research topic in the modern semiconductor industry. And the Drift-Diffusion model is the most simple and widely used model in semiconductor material models. So we use collocation method to solve the Drift-Diffusion models in this paper.This article is divided into five chapters.Chapter1is an introduction. It briefly introduces the origin and the development of the collocation method, as well as its advantages.In Chapter2, we apply the collocation method to solve the Drift-Diffusion models: in which n satisfied with periodic boundary condition, subject to the initial condition and φ is satisfied to the Dirichlet boundary conditionThe collocation scheme and error estimates are obtained.In chapter3, we use the collocation method to solve the two-dimensional Drift-Diffusion models, and obtain the error estimate.Chapter4gives numerical examples of one-dimensional Drift-Diffusion models applying the collocation method. The figures of the approximate solu-tions are presented.Chapter5is a conclusion of the paper.