= Finite element interpolation estimate
On a <shape-regular mesh> in dimensions at most three, nodal piecewise-linear interpolation of $u\in H^2(\Omega)$ satisfies
$$
\|u-I_hu\|_{L^2(\Omega)}+h\|u-I_hu\|_{H^1(\Omega)}
\leq Ch^2\|u\|_{H^2(\Omega)},
$$
where $h$ is the largest element diameter and $C$ is independent of $h$. For homogeneous <Dirichlet boundary conditions>, $I_hu$ 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.
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