A stabilized multi-level method for non-singular finite volume solutions of the stationary 3D Navier–Stokes equations

Abstract

This paper proposes and analyzes a stabilized multi-level finite volume method (FVM) for solving the stationary 3D Navier---Stokes equations by using the lowest equal-order finite element pair without relying on any solution uniqueness condition. This multi-level stabilized FVM consists of solving the nonlinear problem on the coarsest mesh and then performing one Newton correction step on each subsequent mesh, thus only solving a large linear system. An optimal convergence rate for the finite volume approximations of nonsingular solutions is first obtained with the same order as that for the usual finite element solution by using a relationship between the stabilized FVM and a stabilized finite element method. Then the multi-level finite volume approximate solution is shown to have a convergence rate of the same order as that of the stabilized finite volume solution of the stationary Navier---Stokes equations on a fine mesh with an appropriate choice of the mesh size: $${ h_{j} ~ h_{j-1}^{2}, j = 1,\ldots, J}$$ . Finally, numerical results presented validate our theoretical findings.