Two-grid finite-element schemes for the steady Navier-Stokes problem in polyhedra (Q5953290)

Summary: We discretize the steady Navier-Stokes system on a three-dimensional polyhedron by finite element schemes defined on two grids. At the first step, the fully nonlinear problem is solved on a coarse grid, with mesh size \(H\). At the second step, the problem is linearized by substituting into the nonlinear term the velocity \({\mathbf u}_H\) computed at step one, and the linearized problem is solved on a fine grid with mesh size \(h\). This approach is motivated by the fact that the contribution of \({\mathbf u}_H\) to the error analysis is measured in the \(L^3\) norm, and thus, for the lowest-degree elements on a Lipschitz polyhedron, is of the order of \(H^{3/2}\). Hence, an error of the order of \(h\) can be recovered at the second step, provided \(h=H^{3/2}\). When the domain is convex, a similar result can be obtained with \(h=H^2\). Both results are valid in two dimensions.