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February 2021 Fast algorithm based on TT-M FE method for modified Cahn–Hilliard equation
Danxia Wang, Xingxing Wang, Yaqian Li, Hongen Jia
Rocky Mountain J. Math. 51(1): 327-346 (February 2021). DOI: 10.1216/rmj.2021.51.327


We consider numerical methods for solving the modified Cahn–Hilliard equation involving strong nonlinearities. A fast algorithm based on time two-mesh (TT-M) finite element (FE) scheme is proposed to overcome the time-consuming computation of the nonlinear terms. The TT-M FE algorithm includes three main steps: Firstly, a nonlinear FE scheme is solved on a coarse time-mesh τc. Here, the FE method is used for spatial discretization and the implicit second-order 𝜃 scheme (containing both implicit Crank–Nicolson scheme and second-order backward difference method) is used for temporal discretization. Secondly, the Lagrange’s interpolation is used to obtain the interpolation result on the fine time-mesh. Finally, a linearized FE system is solved on a fine time-mesh τ (τ<τc). The stability analysis and priori error estimates are provided in detail. Numerical examples are given to demonstrate the validity of the proposed scheme. The TT-M FE method is compared with the traditional Galerkin FE method and it is evident that the TT-M FE method can save the calculation time.


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Danxia Wang. Xingxing Wang. Yaqian Li. Hongen Jia. "Fast algorithm based on TT-M FE method for modified Cahn–Hilliard equation." Rocky Mountain J. Math. 51 (1) 327 - 346, February 2021.


Received: 2 April 2020; Revised: 12 June 2020; Accepted: 29 June 2020; Published: February 2021
First available in Project Euclid: 28 May 2021

Digital Object Identifier: 10.1216/rmj.2021.51.327

Primary: 65M15 , 65N15

Keywords: CPU time , fast algorithm , modified Cahn–Hilliard equation , priori error estimate , stability , TT-M FE method

Rights: Copyright © 2021 Rocky Mountain Mathematics Consortium


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Vol.51 • No. 1 • February 2021
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