Static two-grid mixed finite-element approximations to the Navier-Stokes equations

TL;DR

This paper introduces a two-grid mixed finite-element scheme for the Navier-Stokes equations, achieving optimal convergence rates and improved accuracy through a postprocessing step on finer meshes.

Contribution

It presents a novel two-grid approach that enhances standard mixed finite-element methods with a postprocessing step, improving convergence rates for Navier-Stokes solutions.

Findings

Optimal convergence rate achieved with suitable coarse mesh size

Postprocessing increases convergence rate by one unit

Numerical experiments confirm theoretical results

Abstract

A two-grid scheme based on mixed finite-element approximations to the incompressible Navier-Stokes equations is introduced and analyzed. In the first level the standard mixed finite-element approximation over a coarse mesh is computed. In the second level the approximation is postprocessed by solving a discrete Oseen-type problem on a finer mesh. The two-level method is optimal in the sense that, when a suitable value of the coarse mesh diameter is chosen, it has the rate of convergence of the standard mixed finite-element method over the fine mesh. Alternatively, it can be seen as a postprocessed method in which the rate of convergence is increased by one unit with respect to the coarse mesh. The analysis takes into account the loss of regularity at initial time of the solution of the Navier-Stokes equations in absence of nonlocal compatibility conditions. Some numerical experiments are shown.