Efficient Finite Element Two-grid Algorithm For Semiconductor Device Problem

In general,semiconductor devices,as the cornerstone of national economic development and national security,can be applied in semiconductor chips,and extensively used in information and communication apparatus,various electron-ic products,new energy industries,and even military equipment.In that case,constructing an algorithm that maintains the reliability of numerical simulation of semiconductors and the optimum accuracy of numerical solution as well as re-duces requirements for calculation time and storages has become a huge challenge for researching modern semiconductor technology.Construction of the two-grid algorithm and analysis of numerical results for semiconductor equations have been studied under the discretization of several types of finite element methods in this paper.According to the study,applying the algorithm to solve the system of nonlinear equations contributes to reducing workload,saving calculation time and obtaining the numerical solution with same order error as the solution using the direct finite element method.First of all,the two-grid method for semiconductor device problem by mixed finite element method and finite element method is studied.The electronic po-tential equation is discretized using the mixed finite element method which also solves the electric field intensity with the consideration of its importance.More-over,a standard finite element method is applied in two concentration equations.Specifically,we prove the L^qerror estimate of the finite element numerical solution based on the property of the ellipse projection operator and the theory of duality argument.Then,a two-grid algorithm is proposed in accordance with iterative operating using Newton's method.The main step is to solve the original nonlinear equations on the coarse grid,and then to deal with the linearized equations on fine grids based on the numerical solutions of coarse grids.The result shows that the proposed method can achieve the optimal approximation as long as the coarse and fine grid sizes satisfy H=O(h^1/2).After that,the efficiency of the two-grid algorithm is verified by the numerical experiment.Secondly,the two-grid method for the problem by mixed finite element method and characteristics finite element method is studied.As a matter of fact,the concentration equations are advection-dominated.Since the discretization of the standard finite element method may result in unacceptable numerical diffusion or non-physical oscillation,the characteristic finite element method is introduced and used to discretize the concentration equations.To begin with,we prove the L^qerror estimate of the mixed finite element-characteristic finite element numerical solution.Next,a two-grid algorithm is constructed with the presentation of error estimate for the numerical solution of the method.In the end,a numerical example is given to verify the reliability and effectiveness of the two-grid method.It can be observed that the concentration equations that are discretized by the mixed finite element method can not only approximate the concentration functions and their flux functions simultaneously,but also maintains the conservation of mass for each element.Hence,the characteristic mixed finite element method is further studied to discretize the concentration equations.By giving the finite element full-discrete scheme and the L⁴error estimate of the numerical solution for the equation system,a two-grid algorithm is designed for this discrete scheme.Further,the error estimate of the two-grid numerical solution can be obtained.In the last part,a three-step two-grid method for semiconductor device prob-lem is studied.To be specific,iterative operation is performed on the fine grid using Newton's method based on the previous two steps.Further,the error estimate of the three-step two-grid numerical solution can be obtained.The results show that sizes of coarse and fine grids satisfy H=O(h^1/4),the three-step two-grid method can still present the finite element method with high precision.After that,the efficiency of the two-grid algorithm is verified by the numerical experiment.