A Multilevel Mesh Independence Principle for the Navier–Stokes Equations

Abstract

Multilevel, finite element discretization methods for the Navier–Stokes equations are considered. In contrast to usual multilevel methods, a superlinear scaling of the consecutive meshwidths $h_J + 1 = \mathcal {O}(h_j^{\alpha (j)} )$ is used. On the coarsest mesh the discretized system is solved, which is small and nonlinear. On each subsequent mesh only one or two Newton correction steps are performed, so that only one or two larger, linear systems are solved. The scalings of the meshwidths that lead to optimal accuracy of the approximate solution in both the $H^1 $- and $L^2 $-norm are investigated.An error analysis of independent interest is also presented for the basic finite element method using the conservation form of the nonlinear term. This formulation leads to a nonlinear system whose nonlinearity is easier to resolve than the systems arising when either the convective or explicitly skew-symmetric corms are used for the nonlinear term.