用于时间依赖Stokes方程的压力鲁棒富集Galerkin方法的全离散分析
Fully discrete analysis of pressure-robust enriched Galerkin methods for the time-dependent Stokes equations
AI总结：
本文针对时间依赖Stokes方程，提出压力鲁棒EG方法的全离散格式，证明其无条件稳定性与最优阶误差估计，数值实验验证其收敛性、局部质量守恒性及黏度鲁棒性，凸显重构算子纳入时间导数项的必要性。
AI中文摘要：
本文针对时间依赖Stokes方程，开展压力鲁棒富集Galerkin（Enriched Galerkin, EG）方法的全离散分析。空间离散采用速度重构算子，该算子保留EG空间的连续分量，同时将其富集不连续分量映射为H(div)空间内的最低阶Raviart-Thomas（$\boldsymbol{\textit{RT}}_0$）子空间，确保严格局部质量守恒与压力鲁棒性。采用向后欧拉法和Crank-Nicolson法构造全离散格式，将该重构算子同时纳入强迫项和离散时间导数项。我们证明了格式的无条件稳定性，并推导了速度和压力的最优阶、参数显式的先验误差估计。特别地，速度估计与连续压力、强迫项的无旋分量无关，不含逆黏度因子，且明确跟踪空间网格尺寸、时间步长和黏度。二维和三维数值实验验证了预测的收敛阶、严格局部质量守恒性以及对黏度的鲁棒性，与非压力鲁棒变体的对比表明，将重构算子纳入离散时间导数项具有重要意义。
英文摘要：
This paper presents a fully discrete analysis of pressure-robust enriched Galerkin (EG) methods for the time-dependent Stokes equations. The spatial discretization employs a velocity reconstruction operator that preserves the continuous component of the EG space while mapping its enriched discontinuous component into the lowest-order Raviart-Thomas ($\mathcal{R}T_0$) subspace of $H(\mathrm{div})$, ensuring strict local mass conservation and pressure-robustness. Fully discrete schemes are constructed using the backward Euler and Crank-Nicolson methods, with the reconstruction incorporated into both the forcing and discrete time-derivative terms. We prove unconditional stability and derive optimal-order, parameter-explicit \textit{a priori} error estimates for both the velocity and pressure. In particular, the velocity estimates are independent of the continuous pressure and the irrotational component of the forcing term, contain no inverse-viscosity factors, and explicitly track the spatial mesh size, time-step size, and viscosity. Numerical experiments in two and three dimensions verify the predicted convergence rates, strict local mass conservation, and robustness with respect to the viscosity, while comparisons with non-pressure-robust variants demonstrate the importance of incorporating the reconstruction into the discrete time-derivative term.
