用于Navier-Stokes方程的grad-div稳定Crank-Nicolson人工压缩性方法的全离散参数鲁棒误差分析
Fully discrete parameter-robust error analysis of a grad-div stabilized Crank-Nicolson artificial compressibility method for the Navier-Stokes equations
AI总结：
针对Navier-Stokes方程，提出带两种grad-div稳定策略的二阶Crank-Nicolson全离散ACM格式，采用Scott-Vogelius有限元对，建立了对ν和ε均鲁棒的最优误差估计，数值实验验证了方法的有效性。
AI中文摘要：
人工压缩性方法（ACM）通过引入与压力时间导数成正比的扰动项（由小正参数ε缩放），松弛Navier-Stokes方程中的不可压缩约束。因此，ACM系统固有包含两个小参数：流体粘度ν和人工压缩性参数ε。尽管现有的ACM时间分析提供了其行为的相关见解，但针对这两个参数均鲁棒的严格全离散误差估计仍是文献中的重大空白。本文提出了二阶Crank-Nicolson全离散ACM格式，并建立了其参数鲁棒最优误差估计。为增强稳定性并确保鲁棒性，我们采用了两种不同的grad-div稳定策略：一种促进速度与压力计算的解耦，另一种保证对ν和ε均鲁棒。空间离散方面，我们采用Scott-Vogelius有限元对，其对解耦和误差分析至关重要。所得的参数一致界对确保格式的长时间精度至关重要，可规避Grönwall引理通常引入的对雷诺数的指数依赖。数值实验被提供以验证理论发现并证明所提方法的效率。
英文摘要：
The artificial compressibility method (ACM) relaxes the incompressibility constraint in the Navier-Stokes equations by introducing a perturbation term proportional to the time derivative of the pressure, scaled by a small positive parameter $\varepsilon$. Consequently, the ACM system inherently involves two small parameters: the fluid viscosity $ν$ and the artificial compressibility parameter $\varepsilon$. While existing temporal analyses of ACM provide insights into its behavior, rigorous fully discrete error estimates that are robust with respect to both parameters remain a significant gap in the literature. In this paper, we propose a second-order Crank-Nicolson fully discrete ACM scheme and establish its parameter-robust optimal error estimates. To enhance stability and ensure robustness, we incorporate two distinct grad-div stabilization strategies: one facilitates the decoupling of velocity and pressure computations, while the other guarantees robustness with respect to both $ν$ and $\varepsilon$. For spatial discretization, we employ the Scott-Vogelius finite element pair, which is crucial for the decoupling and error analysis. The resulting parameter-uniform bounds are crucial for ensuring the long-time accuracy of the scheme, circumventing the exponential dependence on the Reynolds number typically introduced by Grönwall's lemma. Numerical experiments are provided to validate the theoretical findings and demonstrate the efficiency of the proposed methods.
