Discontinuous Galerkin Finite Element Method For The Semiconductor Problem

In 1964, Gummel introduced sequence iteration for this problem [12], and since that time there has been an active development of numerical simulation for semiconductor device. Then, neglecting the influence of temperature, there are many papers. For example, the finite difference method [11], the finite element method [24], the mixed finite element and characteristic finite element methods ([20],[21]), and so on. But following the rapid development of semiconductor device, we should consider the influence of heat-conduction for semiconductor transient problem to avoid the distortion of simulation. Proceeding from production activities, the finite difference method and analysis for the three-dimensional semiconductor problem with heat-conduction were studied in paper [22]. Moving grid finite element method along characteristics was presented in [19]. The results of expending this problem to three-dimension using Galerkin alternating-direction method can be found in paper [14]. In this paper, we use the symmetric interior penalty discontinuous Galerkin (DG, for short) finite element method ([1],[18]) to solve the above problem, and give the analysis of the coupled system.The main motivation for using a discontinuous Galerkin finite element approach for the numerical approximation of the above problem is that DG methods have several interesting features which may be useful in certain applications. Firstly, they easily allow for varying the polynomial order of approximation from one element to the next. Secondly, they allow for very general meshes, including non-conforming meshes, with or without hanging nodes or the need for a mortar space. One can also build stable post-processing into the methods for minimizing oscillations in the presence of high gradients. Thirdly, using the DG method implementation for the hp adaptivity is substantially easier than conforming approaches. Lastly, the methods are locally mass conservative, less numerical diffusive and can treat rough and discontinuous coefficients. From a computational point of view, DG methods are easier to implement than most traditional finite element methods. The trial and test spaces are easier to construct than conforming methods because they are local. This results in a simpler and more efficient implementation. So this method has the extensive application to discontinuous problems.The DG methods naturally accommodate jump conditions and have been employed to solve hyperbolic, parabolic and elliptic problems [8]. For a historical review of DG methods and their applications to elliptic problems we refer to [2]. The approximation properties of the p- and hp-version of the finite element methods were presented in the book [3]. In the papers [18] and [9], continuous in time schemes of primal discontinuous Galerkin methods with interior penalty for miscible displacement problem were proposed.In this paper we use the general meshes. Using the DG method, the gradient of electrostatic potential can be directly computed, which is of the great significance in production activities and greatly improves the accuracy. In order to handle the convection terms in the concentration equations, we use the upwind values of the electron and hole concentrations. In the proof of convergence, we first study the equations separately, assuming the error from the others are known. Then we combine all results to complete the error analysis of the coupled system. For simplicity, we consider the same mesh for the different problems. But different from that for the electrostatic potential equation, we can use integers r₂, r₃, r₄ as the degrees of approximation polynomials for the electron and hole concentration equations, and heat-conduction type equation instead of r₁. Interestingly, the polynomial degrees of approximation spaces for potential and the concentration parts need to be in the same order in order to maintain the convergence of DG applied to the coupled system. That is, r₁/r₂, T₁/t₃, r₂/r₁t and r₃/r₁ need to be bounded.A brief outline of this paper is as follows. In chapter 1, at first the model of the semiconductor problem , some lemmas and continuous at times schemes are introduced. Secondly, we give the properties of Discontinuous Galerkin schemes, and finally obtain the error estimates for the coupled system. In chapter 2, we fist present the extrapolation operator , when we discrete in time for the electrostatic potential. And we give an hypothesis in the proof. Finally, we achieve the error estimates of the fully discrete schemes for the coupled system.Throughout this paper, the symbol C will denote generic positive constant, independent of x, t and all mesh parameters h, r.The symbol ∈ will denote a generic small positive constant. They may take different values at different occurrences.