# American Institute of Mathematical Sciences

doi: 10.3934/era.2020095

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

 1 School of Mathematics and Computational Science, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan 411105, Hunan, China 2 College of Information and Intelligence Science and Technology, Hunan Agricultural University, Changsha 410128, Hunan, China 3 School of Mathematical Science, South China Normal University, Guangzhou 510631, Guangdong, China

* Corresponding author: Yanping Chen

Received  April 2020 Revised  July 2020 Published  September 2020

Fund Project: This work is supported by the State Key Program of National Natural Science Foundation of China (11931003) and National Natural Science Foundation of China (41974133, 11671157, 11971410)

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. Electronic Research Archive, doi: 10.3934/era.2020095
##### References:
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Gummel, A self-consistent iterative scheme for one-dimensional steady-state transistor calculations, IEEE Trans. Electron Devices, 11 (1964), 455-465.  doi: 10.1109/T-ED.1964.15364.  Google Scholar [15] T. I. Seidmann, Time-dependent solutions of a nonlinear system arising in semiconductor theory. Ⅱ. Boundaries and periodicity, Nonlinear Anal., 10 (1986), 491-502.  doi: 10.1016/0362-546X(86)90054-4.  Google Scholar [16] Y. Wang, Y. Chen, Y. Huang and Y. Liu, Two-grid methods for semi-linear elliptic interface problems by immersed finite element methods, Appl. Math. Mech. (English Ed.), 40 (2019), 1657-1676.  doi: 10.1007/s10483-019-2538-7.  Google Scholar [17] M. F. Wheeler, A priori $L_{2}$ error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal., 10 (1973), 723-759.  doi: 10.1137/0710062.  Google Scholar [18] J. Xu, A novel two-grid method for semilinear equations, SIAM J. Sci. 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B, 24 (2019), 387-402.  doi: 10.3934/dcdsb.2018109.  Google Scholar [24] Y. Yuan, Finite difference method and analysis for three-dimensional semiconductor device of heat conduction, Sci. China Ser. A, 39 (1996), 1140-1151.   Google Scholar [25] Y. R. Yuan, A mixed finite element method for the transient behavior of a semiconductor device, Gaoxiao Yingyong Shuxue Xuebao, 7 (1992), 452-463.   Google Scholar [26] Y. R. Yuan, Characteristic finite element method and analysis for numerical simulation of semiconductor devices, Acta Math. Sci. (Chinese), 13 (1993), 241-251.   Google Scholar [27] M. Zlámal, Finite element solution of the fundamental equations of semiconductor devices. I, Math. Comp., 46 (1986), 27-43.  doi: 10.1090/S0025-5718-1986-0815829-6.  Google Scholar

show all references

##### References:
 [1] R. E. Bank, W. M. Coughran, W. Fichtner, E. H. Grosse, D. J. Rose and R. K. Smith, Transient simulation of silicon devices and circuits, IEEE Trans. Computer-Aided Design of Integrated Circuits and Systems, 4 (1985), 436-451.  doi: 10.1109/TCAD.1985.1270142.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] C. Chen, M. Yang and C. Bi, Two-grid methods for finite volume element approximations of nonlinear parabolic equations, J. Comput. Appl. Math., 228 (2009), 123-132.  doi: 10.1016/j.cam.2008.09.001.  Google Scholar [5] Y. Chen and H. Hu, Two-grid method for miscible displacement problem by mixed finite element methods and mixed finite element method of characteristics, Commun. Comput. Phys., 19 (2016), 1503-1528.  doi: 10.4208/cicp.scpde14.46s.  Google Scholar [6] Y. Chen, Y. Huang and D. Yu, A two-grid method for expanded mixed finite-element solution of semilinear reaction-diffusion equations, Internat. J. Numer. Methods Engrg., 57 (2003), 193-209.  doi: 10.1002/nme.668.  Google Scholar [7] Z. Chen, Expanded mixed element methods for linear second-order elliptic problems. Ⅰ, RAIRO Modél. Math. Anal. Numér., 32 (1998), 479-499.  doi: 10.1051/m2an/1998320404791.  Google Scholar [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.  Google Scholar [9] X. Dai and X. Cheng, A two-grid method based on Newton iteration for the Navier–Stokes equations, J. Comput. Appl. Math., 220 (2008), 566-573.  doi: 10.1016/j.cam.2007.09.002.  Google Scholar [10] C. N. Dawson, M. F. Wheeler and C. S. Woodward, A two-grid finite difference scheme for nonlinear parabolic equations, SIAM J. Numer. Anal., 35 (1998), 435-452.  doi: 10.1137/S0036142995293493.  Google Scholar [11] J. Douglas Jr. and J. E. Roberts, Global estimates for mixed methods for second order elliptic equations, Math. Comp., 44 (1985), 39-52.  doi: 10.1090/S0025-5718-1985-0771029-9.  Google Scholar [12] J. Douglas Jr. and T. F. Russell, 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., 19 (1982), 871-885.  doi: 10.1137/0719063.  Google Scholar [13] J. Douglas Jr. and Y. Yuan, Finite difference methods for the transient behavior of a semiconductor device, Mat. Apl. Comput., 6 (1987), 25-37.   Google Scholar [14] H. K. Gummel, A self-consistent iterative scheme for one-dimensional steady-state transistor calculations, IEEE Trans. Electron Devices, 11 (1964), 455-465.  doi: 10.1109/T-ED.1964.15364.  Google Scholar [15] T. I. Seidmann, Time-dependent solutions of a nonlinear system arising in semiconductor theory. Ⅱ. Boundaries and periodicity, Nonlinear Anal., 10 (1986), 491-502.  doi: 10.1016/0362-546X(86)90054-4.  Google Scholar [16] Y. Wang, Y. Chen, Y. Huang and Y. Liu, Two-grid methods for semi-linear elliptic interface problems by immersed finite element methods, Appl. Math. Mech. (English Ed.), 40 (2019), 1657-1676.  doi: 10.1007/s10483-019-2538-7.  Google Scholar [17] M. F. Wheeler, A priori $L_{2}$ error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal., 10 (1973), 723-759.  doi: 10.1137/0710062.  Google Scholar [18] J. Xu, A novel two-grid method for semilinear equations, SIAM J. Sci. Comput., 15 (1994), 231-237.  doi: 10.1137/0915016.  Google Scholar [19] J. Xu, Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal., 33 (1996), 1759-1777.  doi: 10.1137/S0036142992232949.  Google Scholar [20] C. Xu and T. Hou, Superclose analysis of a two-grid finite element scheme for semilinear parabolic integro-differential equations, Electron. Res. Arch., 28 (2020), 897-910.  doi: 10.3934/era.2020047.  Google Scholar [21] Q. Yang, A modified upwind finite volume scheme for semiconductor devices, J. Systems Sci. Math. Sci., 28 (2008), 725-738.   Google Scholar [22] Q. Yang and Y. Yuan, An approximation of semiconductor device by mixed finite element method and characteristics-mixed finite element method, Appl. Math. Comput., 225 (2013), 407-424.  doi: 10.1016/j.amc.2013.09.067.  Google Scholar [23] J. Yu, H. Zheng, F. Shi and R. Zhao, Two-grid finite element method for the stabilization of mixed Stokes-Darcy model, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 387-402.  doi: 10.3934/dcdsb.2018109.  Google Scholar [24] Y. Yuan, Finite difference method and analysis for three-dimensional semiconductor device of heat conduction, Sci. China Ser. A, 39 (1996), 1140-1151.   Google Scholar [25] Y. R. Yuan, A mixed finite element method for the transient behavior of a semiconductor device, Gaoxiao Yingyong Shuxue Xuebao, 7 (1992), 452-463.   Google Scholar [26] Y. R. Yuan, Characteristic finite element method and analysis for numerical simulation of semiconductor devices, Acta Math. Sci. (Chinese), 13 (1993), 241-251.   Google Scholar [27] M. Zlámal, Finite element solution of the fundamental equations of semiconductor devices. I, Math. Comp., 46 (1986), 27-43.  doi: 10.1090/S0025-5718-1986-0815829-6.  Google Scholar
The exact solution $e^{n}$, $h = 1/64$, $n = 100$
The exact solution $p^{n}$, $h = 1/64$, $n = 100$
The exact solution $\psi^{n}$, $h = 1/64$, $n = 100$
Finite element solution $e^{n}_{h}$, $h = 1/64$, $n = 100$
Finite element solution $p^{n}_{h}$, $h = 1/64$, $n = 100$
Finite element solution $\psi^{n}_{h}$, $h = 1/64$, $n = 100$
Two-grid solution $E^{n}_{h}$, $H = 1/8$, $h = 1/64$, $n = 100$
Two-grid solution $P^{n}_{h}$, $H = 1/8$, $h = 1/64$, $n = 100$
Two-grid solution $\Psi^{n}_{h}$, $H = 1/8$, $h = 1/64$, $n = 100$
Order of finite element solution $e^{n}_{h}$, $n = 100$
Order of finite element solution $p^{n}_{h}$, $n = 100$
Order of finite element solution ${\boldsymbol{u}}^{n}_h$, $n = 100$
Order of finite element solution $\psi^{n}_{h}$, $n = 100$
Error and CPU time of the finite element method for $n = 100$
 $h$ $\|e^n-e^n_h\|$ $\|p^n-p^n_h\|$ $\|\psi^n-\psi^n_h\|$ $\|{\boldsymbol{u}}^n-{\boldsymbol{u}}^n_h\|$ CPU time $\frac{1}{4}$ 2.0125e-2 4.1448e-3 4.6324e-4 2.4411e-3 0.4850s $\frac{1}{8}$ 6.3665e-3 1.1304e-3 2.3741e-4 1.2207e-3 1.0107s $\frac{1}{16}$ 1.6748e-3 3.1703e-4 1.1935e-4 6.0937e-4 3.4394s $\frac{1}{32}$ 4.0476e-4 1.1037e-4 5.9754e-5 3.0453e-4 15.1790s $\frac{1}{64}$ 9.3634e-5 5.9437e-5 2.9886e-5 1.5224e-4 74.9759s
 $h$ $\|e^n-e^n_h\|$ $\|p^n-p^n_h\|$ $\|\psi^n-\psi^n_h\|$ $\|{\boldsymbol{u}}^n-{\boldsymbol{u}}^n_h\|$ CPU time $\frac{1}{4}$ 2.0125e-2 4.1448e-3 4.6324e-4 2.4411e-3 0.4850s $\frac{1}{8}$ 6.3665e-3 1.1304e-3 2.3741e-4 1.2207e-3 1.0107s $\frac{1}{16}$ 1.6748e-3 3.1703e-4 1.1935e-4 6.0937e-4 3.4394s $\frac{1}{32}$ 4.0476e-4 1.1037e-4 5.9754e-5 3.0453e-4 15.1790s $\frac{1}{64}$ 9.3634e-5 5.9437e-5 2.9886e-5 1.5224e-4 74.9759s
Error and CPU time of the two-grid method for $n = 100$, $H = \frac{1}{8}$ with different $h = \frac{1}{16}, \frac{1}{32}, \frac{1}{64}$
 $H$ $h$ $\|e^n-E^n_h\|$ $\|p^n-P^n_h\|$ $\|\psi^{n}-\Psi^{n}_{h}\|$ $\|{\boldsymbol{u}}^n-{\boldsymbol{U}}^n_h\|$ CPU time $\frac{1}{8}$ $\frac{1}{16}$ 1.6698e-3 3.3035e-4 1.1935e-4 6.0937e-4 1.0793s $\frac{1}{8}$ $\frac{1}{32}$ 4.0302e-4 1.1523e-4 5.9754e-5 3.0453e-4 4.1308s $\frac{1}{8}$ $\frac{1}{64}$ 9.7872e-5 6.4727e-5 2.9886e-5 1.5224e-4 20.4816s
 $H$ $h$ $\|e^n-E^n_h\|$ $\|p^n-P^n_h\|$ $\|\psi^{n}-\Psi^{n}_{h}\|$ $\|{\boldsymbol{u}}^n-{\boldsymbol{U}}^n_h\|$ CPU time $\frac{1}{8}$ $\frac{1}{16}$ 1.6698e-3 3.3035e-4 1.1935e-4 6.0937e-4 1.0793s $\frac{1}{8}$ $\frac{1}{32}$ 4.0302e-4 1.1523e-4 5.9754e-5 3.0453e-4 4.1308s $\frac{1}{8}$ $\frac{1}{64}$ 9.7872e-5 6.4727e-5 2.9886e-5 1.5224e-4 20.4816s
Error and CPU time of the two-grid method for $n = 100$, and $h = H^{2} = \frac{1}{4}, \frac{1}{16}, \frac{1}{64}$
 $H$ $h$ $\|e^n-E^n_h\|$ $\|p^n-P^n_h\|$ $\|\psi^{n}-\Psi^{n}_{h}\|$ $\|{\boldsymbol{u}}^n-{\boldsymbol{U}}^n_h\|$ CPU time $\frac{1}{2}$ $\frac{1}{4}$ 2.0132e-2 4.1526e-3 4.6324e-4 2.4411e-3 0.1774s $\frac{1}{4}$ $\frac{1}{16}$ 1.6698e-3 3.3035e-4 1.1935e-4 6.0937e-4 1.0310s $\frac{1}{8}$ $\frac{1}{64}$ 9.7872e-5 6.4727e-5 2.9886e-5 1.5224e-4 20.4816s
 $H$ $h$ $\|e^n-E^n_h\|$ $\|p^n-P^n_h\|$ $\|\psi^{n}-\Psi^{n}_{h}\|$ $\|{\boldsymbol{u}}^n-{\boldsymbol{U}}^n_h\|$ CPU time $\frac{1}{2}$ $\frac{1}{4}$ 2.0132e-2 4.1526e-3 4.6324e-4 2.4411e-3 0.1774s $\frac{1}{4}$ $\frac{1}{16}$ 1.6698e-3 3.3035e-4 1.1935e-4 6.0937e-4 1.0310s $\frac{1}{8}$ $\frac{1}{64}$ 9.7872e-5 6.4727e-5 2.9886e-5 1.5224e-4 20.4816s
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