P 0-approximation of $${\Delta(H_0^2 \cap W^{3,\infty})}$$ Δ ( H 0 2 ∩ W 3 , ∞ ) on square grids based on interior squares

Abstract

We will approximate a function in \({\Delta(H_0^2 \cap W^{3,\infty})}\) with a piecewise constant function on the square grids, whose degrees of freedom are based on the interior squares only. In spite of the slight lack of degrees of freedom, the approximation error is analyzed to be O(h) in L2-norm, which is the optimal order for the piecewise-constant interpolation. The result in the paper is relevant to the finite element approximation for the divergence-free velocity in the Stokes problem with the locally divergence-free P1-nonconforming space on the square grids.