\(P_0\)-approximation of \(\Delta(H_0^2\cap W^{3,\infty})\) on square grids based on interior squares (Q2452282)

The approximation error for a function \(\Delta (H_0^2\cap W^{3,\infty})\) with a piecewise constant function on the square grids is analyzed. This type of approximations occur in finite element discretization of the divergence-free velocity in the Stokes problem with the locally divergence-free \(P_1\)-nonconforming space on the square grids. The error is of order \(\mathcal{O}(h)\) in the \(L^2\)-norm, which is optimal for the piecewise-constant interpolation. The standard local error analysis, such as Bramble-Hilbert's lemma is not sufficient due to the slight lack of degrees of freedom. Therefore, the stronger regularity \(W^{3,\infty}\) is used, compared to \(H^2_0\) for a global analysis in the entire domain.