Development of a Locally Analytic Prolongation Operator in the Two-Grid, Three-Level Method for the Navier-Stokes Equations

In this article, two three-level methods employing the same prolongation operator are proposed for efficiently solving the incompressible Navier-Stokes equations in a two-grid system. Each method involves solving one smaller system of nonlinear equations in the coarse mesh. The chosen Newton- or Oseen-type linearized momentum equations along with a correction step are solved only once on the fine mesh. Within the three-level framework, the locally analytic prolongation operator needed to bridge the convergent Navier-Stokes solutions obtained at the coarse mesh and the interpolated velocities at the fine mesh is developed to improve the prediction quality. To increase prediction accuracy, the linearized momentum equations are discretized within the alternating direction implicit context using our previously developed nodally exact convection-diffusion-reaction finite-difference scheme. Two proposed three-level methods are rigorously assessed in terms of simulated accuracy, nonlinear convergence rate, and elapsed CPU time.