Finite element interpolation estimate

On a shape-regular mesh in dimensions at most three, nodal piecewise-linear interpolation of $u\in H^2(\Omega )$ satisfieswhere $h$ is the largest element diameter and $C$ is independent of $h$. For homogeneous Dirichlet boundary conditions, $I_{h}u$ belongs to the conforming finite element space. Scaling each element to a reference element gives the estimate; shape regularity bounds the scaling constants, while subtraction of an affine function leaves an error controlled by the second derivatives.