An Ultra-Weak Discontinuous Galerkin Method For Drift-Diffusion Model Of Semiconductor Problem

In this paper,the ultra-weak discontinuous Galerkin(UWDG)method is considered to numerically simulate the drift-diffusion(DD)model of semiconductor problems.The corresponding error analysis of the UWDG method of the DD model is carried out.The UWDG method refers to a discontinuous Galerkin(DG)method for the partial integral formula by transfering all spatial derivatives from solutions to test functions in the weak formulations.The UWDG method,like local discontinuous Galerkin(LDG)method,is commonly used for partial differential equations with time variables and higher-order spatial derivatives.Compared with the LDG method,the main advantage of the UWDG method is that it does not need auxiliary variables,and is more intuitive and concise in form,thereby reducing memory and computational costs.Moreover,the UWDG method has the characteristics of h-p adaptability and local communication of the traditional DG method.The DD model is suitable for micron-level device simulation,and is a model used to calculate the distribution of potential and electron concentration in semiconductors,which is obtained by coupling two common partial differential equations with higher-order spatial derivatives and time variables,the drift-diffusion equation and the Poisson equation.The focus of this article is error analysis of our format.We need to choose the appropriate projection to analyze the error.Young’s inequality and Schwartz’s inequality are also used several times during the error analysis.The error analysis in this paper finally obtains the order of k-order error in the sense of L2 norm.In the process of numerical simulation,the numerical results obtained by the UWDG method are compared with the results obtained by the LDG method and another DG method that uses only one partial integral for all terms containing spatial derivatives.The results of the two basically coincide,which verifies the excellent ability of the UWDG method in simulating the drift-diffusion model of the semiconductor problem.At the same time,numerical result verifies the stability of the UWDG method.This article is divided into six chapters.In the first chapter,the origin and development of the DD model and UWDG method are briefly introduced,and the overall context and content framework of the article are summarized.In the second chapter,the basics necessary for this article are presented.including the basic symbols and some important properties covered in the article.In the third chapter,two DG methods for the convective diffusion equation are described and the advantages of the UWDG method are illustrated with a simple numerical example.In the fourth chapter.the scheme and error analysis of the UWDG method for solving one-dimensional DD models are given.This chapter is the focus of this article.In the fifth chapter,the study is numerically simulated,the required parameters are explained and the results of the numerical simulation are explained.Finally,the full text is summarized and looked forward to in the sixth chapter.