### Title

Interpolation error estimate

### Description

These are my notes for coding the problem of interpolation error estimate.

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## $L^\infty$-norm error constant estimation for the linear Lagrange interpolation

Let $K$ be a triangular domain with vertices $P_1, P_2, P_3$. Let $\Pi_h$ be the Lagrange interpolation over triangle $K$, such that for $u \in H^2(K)$,

$$(\Pi_h u -u )(p) = 0,\quad p=P_1, P_2, P_3\:.$$

Let us consider the following error estimation for $\Pi_h$ with error constant $C$,

$$|u-\Pi_h u|{\infty} \le C |u|\:.$$

Here, $|\cdot|{2,K}$ and $|\cdot|\infty$ denote the $H^2$-seminorm and the $L^\infty$ norm of a function, respectively.

The determination of $C$ is to solve an optimization problem under the constraint condition with $L^\infty$ norm.

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