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About the project
This project demonstrates the method of estimating the upper bound of the $L^\infty$norm of the linear Lagrange interpolation error.
$L^\infty$norm error constant estimation for the linear Lagrange interpolation
Let $K$ be a triangular domain with vertices $P_1, P_2, P_3$. Let $\Pi$ be the Lagrange interpolation over triangle $K$, such that for $u \in H^2(K)$, $(\Pi^L u u )(p) = 0,\quad p=P_1, P_2, P_3$ .
Let us consider the following error estimation for $\Pi^L$ with error constant $C^L(K)$,
$$  u  \Pi^L u {\infty,K} \leq C^L(K) u . $$
Here, $ . {2,K}$ and $. $ denote the $H^2$seminorm and the $L^\infty$ norm of a function, respectively.
Content Description
The following notebooks are contained in the project.

Maximum Norm Error Estimation
 Here, the framework of the concept and algorithm is explained.

Comparison of the upper bounds for different mesh size
 In this notebook, for a right isosceles triangle $K$, the upper bound for $C^L(K)$ is obtained using uniform triangulation. It is demonstrated how the upper bound changes as the mesh is refined.

Lower Bounds of the Constant
 In this notebook, the lower bound of the constant is also determined to verify sharpness.

Contour Lines
 The upper bound of the constant for a triangle with vertices $P_1(0,0)$, $P_2(1,0)$ and $p_3(x,y)$. As the third vertex varies in the first and second quadrant of the $(x,y)$plane, the upper bound of the constant also varies. Here, the contour lines of the upper bound are graphed.
% Edited Shirley Mae Galindo 09/11/2021
About the directory
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