Numerical methods for Navier-Stokes equation and elasticity problem lead to a class of $2\times 2$ block structure linear system. Based on a general splitting for the (1, 1) block of coefficient matrix and the Uzawa method, we consider a general Uzawa-type method for solving this $2\times 2$ block structure linear system in this paper. With different choices of splitting matrix, the proposed Uzawa-type method includes some existing Uzawa-type methods as its special cases. The convergence and semi-convergence of the proposed method for solving nonsingular and singular problems are studied, respectively. When the splitting matrix is used as a preconditioner, we analyze the spectral properties of the preconditioned matrix. Numerical results are presented to verify the efficiency of the proposed method.

Navier-Stokes 方程和弹性问题的数值方法导致一类 $2\times 2$块结构线性系统。基于系数矩阵的 (1, 1) 块的一般分裂和 Uzawa 方法，我们考虑了一个一般 Uzawa 类型的方法来解决这个问题$2\times 2$本文采用块结构线性系统。随着分裂矩阵的选择不同，所提出的 Uzawa 型方法包括一些现有的 Uzawa 型方法作为其特例。分别研究了所提出的解决非奇异问题和奇异问题的方法的收敛性和半收敛性。当分裂矩阵用作预处理器时，我们分析了预处理矩阵的光谱特性。给出了数值结果以验证所提出方法的有效性。