Abstract
The main purpose of this paper is to solve the viscous Cahn-Hilliard equation via a fast algorithm based on the two time-mesh (TT-M) finite element (FE) method to ease the problem caused by strong nonlinearities. The TT-M FE algorithm includes the following main computing steps. First, a nonlinear FE method is applied 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 and second-order backward difference) is used for temporal discretization. Second, based on the chosen initial iterative value, a linearized FE system on time fine mesh is solved, where some useful coarse numerical solutions are found by Lagrange’s interpolation formula. The analysis for both stability and a priori error estimates is made in detail. Numerical examples are given to demonstrate the validity of the proposed algorithm. Our algorithm is compared with the traditional Galerkin FE method and it is evident that our fast algorithm can save computational time.
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Acknowledgements
This work was supported by the Research Project Supported by Shanxi Scholarship Council of China (No.2021-029), the Key Research and Development (R&D) Projects of Shanxi Province (No. 201903D121038), the Natural Science Foundation of Shanxi Province (Nos. 201801D121016, 201901D111123).
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Wang, D., Li, Y., Wang, X. et al. Fast algorithm for viscous Cahn-Hilliard equation. Front. Math. China 17, 689–713 (2022). https://doi.org/10.1007/s11464-021-0974-x
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DOI: https://doi.org/10.1007/s11464-021-0974-x
Keywords
- Fast algorithm
- two time-mesh (TT-M) finite element (FE) method
- viscous Cahn-Hilliard equation
- stability
- CPU time