In this paper, a new multiplicative Schwarz method is presented for solving a system arising from the discretization of the nonselfadjoint elliptic equations. In the implementation, we apply the proposed Schwarz method as a Field-of-Value (FOV) equivalent preconditioner which is accelerated with the GMRES iterative solver. By employing a strengthened Cauchy–Schwarz inequality and a stable multilevel decomposition under a new norm, we obtain the optimal convergence theory by choosing the parameters in the Schwarz operator appropriately. It shows that the lower and upper bounds for the spectrum of the preconditioned Schwarz operator are bounded independently of the fine mesh size, the number of subdomains and mesh levels. Some numerical results are reported to verify the theory in terms of optimality and scalability. Moreover, numerical comparisons show that the proposed method is competitive with the classical multiplicative Schwarz algorithm for solving the convection–diffusion equations.

在本文中，提出了一种新的乘法 Schwarz 方法来求解由非自伴随椭圆方程离散化产生的系统。在实现中，我们将建议的 Schwarz 方法用作值域 (FOV) 等效预处理器，该预处理器通过 GMRES 迭代求解器进行加速。通过在新规范下采用加强的Cauchy-Schwarz不等式和稳定的多级分解，我们通过适当地选择Schwarz算子中的参数来获得最佳收敛理论。它表明预处理的 Schwarz 算子的频谱的下限和上限与细网格大小、子域数量和网格级别无关。报告了一些数值结果以在最优性和可扩展性方面验证该理论。而且，