Parallel multiplicative Schwarz preconditioner for solving nonselfadjoint elliptic problems

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.