### Electronic Research Archive

2021, Issue 1: 1859-1880. doi: 10.3934/era.2020095
Special Issues

# Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method

• Received: 01 April 2020 Revised: 01 July 2020 Published: 16 September 2020
• Primary: 35M13; Secondary: 65M12

• The mathematical model of a semiconductor device is described by a coupled system of three quasilinear partial differential equations. The mixed finite element method is presented for the approximation of the electrostatic potential equation, and the characteristics finite element method is used for the concentration equations. First, we estimate the mixed finite element and the characteristics finite element method solution in the sense of the $L^q$ norm. To linearize the full discrete scheme of the problem, we present an efficient two-grid method based on the idea of Newton iteration. The two-grid algorithm is to solve the nonlinear coupled equations on the coarse grid and then solve the linear equations on the fine grid. Moreover, we obtain the $L^{q}$ error estimates for this algorithm. It is shown that a mesh size satisfies $H = O(h^{1/2})$ and the two-grid method still achieves asymptotically optimal approximations. Finally, the numerical experiment is given to illustrate the theoretical results.

Citation: Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method[J]. Electronic Research Archive, 2021, 29(1): 1859-1880. doi: 10.3934/era.2020095

### Related Papers:

• The mathematical model of a semiconductor device is described by a coupled system of three quasilinear partial differential equations. The mixed finite element method is presented for the approximation of the electrostatic potential equation, and the characteristics finite element method is used for the concentration equations. First, we estimate the mixed finite element and the characteristics finite element method solution in the sense of the $L^q$ norm. To linearize the full discrete scheme of the problem, we present an efficient two-grid method based on the idea of Newton iteration. The two-grid algorithm is to solve the nonlinear coupled equations on the coarse grid and then solve the linear equations on the fine grid. Moreover, we obtain the $L^{q}$ error estimates for this algorithm. It is shown that a mesh size satisfies $H = O(h^{1/2})$ and the two-grid method still achieves asymptotically optimal approximations. Finally, the numerical experiment is given to illustrate the theoretical results.

 [1] Transient simulation of silicon devices and circuits. IEEE Trans. Computer-Aided Design of Integrated Circuits and Systems (1985) 4: 436-451. [2] S. C. Brenner and L. Ridgway Scott, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, 15, Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0 [3] F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge, 8 (1974), 129–151. doi: 10.1051/m2an/197408R201291 [4] Two-grid methods for finite volume element approximations of nonlinear parabolic equations. J. Comput. Appl. Math. (2009) 228: 123-132. [5] Two-grid method for miscible displacement problem by mixed finite element methods and mixed finite element method of characteristics. Commun. Comput. Phys. (2016) 19: 1503-1528. [6] A two-grid method for expanded mixed finite-element solution of semilinear reaction-diffusion equations. Internat. J. Numer. Methods Engrg. (2003) 57: 193-209. [7] Expanded mixed element methods for linear second-order elliptic problems. Ⅰ. RAIRO Modél. Math. Anal. Numér. (1998) 32: 479-499. [8] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Studies in Mathematics and its Applications, 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. doi: 10.1137/1.9780898719208 [9] A two-grid method based on Newton iteration for the Navier–Stokes equations. J. Comput. Appl. Math. (2008) 220: 566-573. [10] A two-grid finite difference scheme for nonlinear parabolic equations. SIAM J. Numer. Anal. (1998) 35: 435-452. [11] Global estimates for mixed methods for second order elliptic equations. Math. Comp. (1985) 44: 39-52. [12] Numerical methods for convention-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures. SIAM J. Numer. Anal. (1982) 19: 871-885. [13] Finite difference methods for the transient behavior of a semiconductor device. Mat. Apl. Comput. (1987) 6: 25-37. [14] A self-consistent iterative scheme for one-dimensional steady-state transistor calculations. IEEE Trans. Electron Devices (1964) 11: 455-465. [15] Time-dependent solutions of a nonlinear system arising in semiconductor theory. Ⅱ. Boundaries and periodicity. Nonlinear Anal. (1986) 10: 491-502. [16] Two-grid methods for semi-linear elliptic interface problems by immersed finite element methods. Appl. Math. Mech. (English Ed.) (2019) 40: 1657-1676. [17] A priori $L_{2}$ error estimates for Galerkin approximations to parabolic partial differential equations. SIAM J. Numer. Anal. (1973) 10: 723-759. [18] A novel two-grid method for semilinear equations. SIAM J. Sci. Comput. (1994) 15: 231-237. [19] Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal. (1996) 33: 1759-1777. [20] Superclose analysis of a two-grid finite element scheme for semilinear parabolic integro-differential equations. Electron. Res. Arch. (2020) 28: 897-910. [21] A modified upwind finite volume scheme for semiconductor devices. J. Systems Sci. Math. Sci. (2008) 28: 725-738. [22] An approximation of semiconductor device by mixed finite element method and characteristics-mixed finite element method. Appl. Math. Comput. (2013) 225: 407-424. [23] Two-grid finite element method for the stabilization of mixed Stokes-Darcy model. Discrete Contin. Dyn. Syst. Ser. B (2019) 24: 387-402. [24] Finite difference method and analysis for three-dimensional semiconductor device of heat conduction. Sci. China Ser. A (1996) 39: 1140-1151. [25] A mixed finite element method for the transient behavior of a semiconductor device. Gaoxiao Yingyong Shuxue Xuebao (1992) 7: 452-463. [26] Characteristic finite element method and analysis for numerical simulation of semiconductor devices. Acta Math. Sci. (Chinese) (1993) 13: 241-251. [27] Finite element solution of the fundamental equations of semiconductor devices. I. Math. Comp. (1986) 46: 27-43.
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

1.833 0.8

Article outline

## Figures and Tables

Figures(13)  /  Tables(3)

• On This Site