### Electronic Research Archive

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

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沈阳化工大学材料科学与工程学院 沈阳 110142

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