Optimal error bounds for two-grid schemes applied to the Navier-Stokes equations
Provides rigorous error analysis for a computationally efficient two-grid method for incompressible Navier-Stokes equations, important for numerical analysts and engineers.
The paper proves optimal error bounds for two-grid mixed-finite element schemes for the Navier-Stokes equations, accounting for low regularity at initial times. The method achieves optimal convergence rates despite the lack of compatibility conditions.
We consider two-grid mixed-finite element schemes for the spatial discretization of the incompressible Navier-Stokes equations. A standard mixed-finite element method is applied over the coarse grid to approximate the nonlinear Navier-Stokes equations while a linear evolutionary problem is solved over the fine grid. The previously computed Galerkin approximation to the velocity is used to linearize the convective term. For the analysis we take into account the lack of regularity of the solutions of the Navier-Stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data. Optimal error bounds are obtained.