A novel two-grid Crank-Nicolson mixed finite element method for nonlinear fourth-order sin-Gordon equation

Yanjie Zhou; Xianxiang Leng; Yuejie Li; Qiuxiang Deng; Zhendong Luo; Yanjie Zhou; Xianxiang Leng; Yuejie Li; Qiuxiang Deng; Zhendong Luo

doi:10.3934/math.20241515

AIMS Mathematics

2024, Volume 9, Issue 11: 31470-31494. doi: 10.3934/math.20241515

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Research article

A novel two-grid Crank-Nicolson mixed finite element method for nonlinear fourth-order sin-Gordon equation

1.
School of Mathematics and Statistics, Beijing Technology and Business University, Beijing 100048, China
2.
Department of Mathematics and Computer Engineering, Ordos Institute of Technology, Ordos 017000, China
3.
Academician Expert Workstation, Hunan Sany Polytechnic College, Changsha 410129, China

Received: 18 September 2024 Revised: 28 October 2024 Accepted: 31 October 2024 Published: 05 November 2024
MSC : 65M15, 65N12, 65N35

A new nonlinear fourth-order sin-Gordon equation with actual physical background is first created. Then, by introducing an auxiliary function, the nonlinear fourth-order sin-Gordon equation is decomposed into the nonlinear system of equations of second-order derivatives of spatial variables. Subsequently, the time derivative is discretized by using the Crank-Nicolson (CN) scheme to construct a new time semi-discretized mixed CN (TSDMCN) scheme. Thereafter, the spatial variables in the TSDMCN scheme are discretized by using a two-grid mixed finite element (MFE) method to construct a new two-grid CN MFE (TGCNMFE) method with unconditional stability and second-order time precision, which consists of a system of nonlinear MFE equations defined on coarser grids and a system of linear MFE equations defined on finer grids with sufficiently high precision, and is very easy to solve. The existence, stability, and error estimates of the TSDMCN and TGCNMFE solutions are strictly proved theoretically, and the superiorities of the TGCNMFE method and the correctness of theoretical results are verified by two sets of numerical experiments.
Citation: Yanjie Zhou, Xianxiang Leng, Yuejie Li, Qiuxiang Deng, Zhendong Luo. A novel two-grid Crank-Nicolson mixed finite element method for nonlinear fourth-order sin-Gordon equation[J]. AIMS Mathematics, 2024, 9(11): 31470-31494. doi: 10.3934/math.20241515

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Abstract

A new nonlinear fourth-order sin-Gordon equation with actual physical background is first created. Then, by introducing an auxiliary function, the nonlinear fourth-order sin-Gordon equation is decomposed into the nonlinear system of equations of second-order derivatives of spatial variables. Subsequently, the time derivative is discretized by using the Crank-Nicolson (CN) scheme to construct a new time semi-discretized mixed CN (TSDMCN) scheme. Thereafter, the spatial variables in the TSDMCN scheme are discretized by using a two-grid mixed finite element (MFE) method to construct a new two-grid CN MFE (TGCNMFE) method with unconditional stability and second-order time precision, which consists of a system of nonlinear MFE equations defined on coarser grids and a system of linear MFE equations defined on finer grids with sufficiently high precision, and is very easy to solve. The existence, stability, and error estimates of the TSDMCN and TGCNMFE solutions are strictly proved theoretically, and the superiorities of the TGCNMFE method and the correctness of theoretical results are verified by two sets of numerical experiments.

References

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