Navier-Stokes方程的一阶IMEX-SAV有限元格式的无条件最优误差估计
Unconditionally Optimal Error Estimates of a First-Order IMEX-SAV Finite Element Scheme for the Navier-Stokes Equations
摘要: 基于指数函数的标量辅助变量(SAV)方法，求解不可压缩Navier-Stokes方程的一阶Euler隐式–显式(IMEX)时间半离散数值格式，其优点在于：所构造的时间离散格式是无条件稳定的，并且在每个时间步上，仅需求解Stokes问题。借助SAV方法，本文构造了求解Navier-Stokes方程的一阶Euler隐式–显式有限元全离散格式，理论上证明了所构造的全离散格式是无条件稳定的。基于误差分裂技巧，理论上得到了速度和压力的无条件最优误差估计。最后，给出数值算例验证理论分析结果。
Abstract: A first-order Euler implicit-explicit (IMEX) temporal semi-discrete numerical scheme for solving the incompressible Navier-Stokes equations based on the scalar auxiliary variable (SAV) method was proposed. Its advantages lie in two aspects: the constructed temporal discrete scheme is unconditionally stable, and only the Stokes problem needs to be solved at each time step. By virtue of the SAV method, this paper develops a first-order Euler IMEX fully discrete finite element scheme for the Navier-Stokes equations. Theoretically, it is proved that the constructed fully discrete scheme is unconditionally stable. Based on the error splitting technique, the unconditionally optimal error estimates for velocity and pressure are derived theoretically. Finally, numerical examples are provided to verify the theoretical analysis results.
文章引用：李健. Navier-Stokes方程的一阶IMEX-SAV有限元格式的无条件最优误差估计[J]. 应用数学进展, 2025, 14(12): 453-467. https://doi.org/10.12677/aam.2025.1412521

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