通过最小二乘弱伽辽金方法求解斯托克斯方程
Solving the Stokes Equations via a Least Squares Weak Galerkin Method
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中文总结 AI 辅助
研究在任意多边形和多面体网格上求解斯托克斯方程，提出最小二乘弱伽辽金有限元方法，利用离散弱导数适应复杂几何，绕过LBB条件转化为对称正定系统，推导最优阶误差估计，数值实验验证了收敛率。
中文摘要 AI 辅助
我们提出了一种最小二乘弱伽辽金（LS-WG）有限元方法，用于在任意多边形和多面体网格上求解斯托克斯方程。该框架利用不连续多项式空间上的离散弱导数，自然地适应复杂的区域几何形状和一般划分。关键的是，这种最小二乘公式绕过了传统的下-上（LBB）相容性条件，将标准的不定鞍点问题转化为一个本质上对称正定（SPD）的离散线性系统。我们建立了数值格式的适定性，并在定制的离散能量范数中严格推导了最优阶误差估计。具体而言，当速度场采用次数$k \ge 1$的多项式且压力采用次数$k - 1$的多项式时，我们证明了离散投影误差的收敛率为$\mathcal{O}(h^k)$，全局逼近误差的收敛率为$\mathcal{O}(h^{k - 1})$。大量数值实验证实了这些理论收敛率，证明了该方法的鲁棒性、几何灵活性和整体效率。
英文摘要
We present a least-squares weak Galerkin (LS-WG) finite element method for solving the Stokes equations on arbitrary polygonal and polyhedral meshes. By utilizing discrete weak derivatives on discontinuous polynomial spaces, the proposed framework naturally accommodates complex domain geometries and general partitions. Crucially, this least-squares formulation bypasses the traditional inf-sup (LBB) compatibility condition, transforming the standard indefinite saddle-point problem into an inherently symmetric and positive definite (SPD) discrete linear system. We establish the well-posedness of the numerical scheme and rigorously derive optimal-order error estimates in a custom discrete energy norm. Specifically, we prove convergence rates of $\mathcal{O}(h^k)$ for the discrete projection error and $\mathcal{O}(h^{k-1})$ for the global approximation error when employing polynomials of degree $k \ge 1$ for the velocity field and $k-1$ for the pressure. Extensive numerical experiments confirm these theoretical convergence rates, demonstrating the method's robustness, geometric flexibility, and overall efficiency.
