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A second-order backward differentiation formula for the numerical solution of Cahn–Hilliard–Hele–Shaw system

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Abstract

In this paper, a second-order, unconditionally stable numerical scheme for the Cahn–Hilliard–Hele–Shaw system is constructed, which combines the backward differentiation formula in time and the mixed finite element method in space. Based on the Lagrange multiplier approach, we develop the linear discretization for the nonlinear term, which is extremely efficient. By rigorous proofs and calculations, we prove the unique solvability of the numerical solution, the unconditional stability and the error estimates of the proposed scheme. Furthermore, some numerical tests are given on the temporal and spatial convergence rates, spinodal decomposition and energy decay, which are consistent with the results of the theoretical analysis.

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Funding

This work was supported by the Research Project Supported by Shanxi Scholarship Council of China(No.2021-029) and Shanxi Provincial International Cooperation Base and Platform Project (202104041101019).

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Author notes

  1. Yuanjing Nie and Danxia Wang contributed equally to this work.

Authors and Affiliations

  1. College of Mathematics, Taiyuan University of Technology, Wanbolin, Taiyuan, 030024, Shanxi, China

    Xianxia Wang & Danxia Wang

  2. College of Mathematics, Taiyuan University of Technology, Yuci, Jinzhong, 030060, Shanxi, China

    Yuanjing Nie

Authors
  1. Xianxia Wang
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  2. Yuanjing Nie
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  3. Danxia Wang
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Correspondence to Danxia Wang.

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Wang, X., Nie, Y. & Wang, D. A second-order backward differentiation formula for the numerical solution of Cahn–Hilliard–Hele–Shaw system. Comp. Appl. Math. 42, 135 (2023). https://doi.org/10.1007/s40314-023-02280-3

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  • DOI: https://doi.org/10.1007/s40314-023-02280-3

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