Anderson acceleration of Picard/Newton iteration based on grad-div stabilization for the Smagorinsky model

In this paper, we consider Anderson acceleration of one- and two-level Picard/Newton iterations based on the grad-div stabilization for the stationary Smagorinsky model at high Reynolds number. First, based on the grad-div stabilization, we propose the Anderson-accelerated Picard iteration, and then we introduce the Newton iteration at the end of iteration to accelerate convergence. Second, to reduce the computational cost, we consider a two-level algorithm, i.e., we use the previous Anderson acceleration of Picard/Newton iteration for the Smagorinsky model on a coarse mesh and then solve a generalized Stokes problem on fine mesh by the grad-div stabilized Picard iteration. The proposed algorithm not only improves convergence and reduces computational costs but also enhances the capability to simulate the fluid flow with high Reynolds number. Several numerical experiments have been conducted to demonstrate the numerical performance of the proposed algorithms.