Parallel defect-correction methods for incompressible flows with friction boundary conditions

Abstract

We study three parallel defect-correction methods based on finite element approximations for the incompressible Navier–Stokes problem with friction boundary conditions and high Reynolds numbers in this work, where a fully overlapping domain decomposition is considered for parallelization. In the proposed methods, with a global multiscale grid that builds a fine grid around its own subdomain and coarse elsewhere, we iteratively solve an artificial viscosity nonlinear variational inequality problem in a defect step, and then compute the residual by the linearized variational inequality problems in the r-step corrections. The studied methods are easy to implement on the basis of the existing Navier–Stokes solver and possess less communication complexity. We provide a rigorously theoretical derivation for the error estimates of the one-step correction solutions from the proposed methods under some stable conditions, and derive scalings of the algorithmic parameters. We demonstrate by a series of numerical experiments that the velocity and pressure errors computed by our parallel defect-correction methods are comparable to those of the standard defect-correction method, while our present methods reduce the computational cost.