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About the project
This project demonstrates the method of estimating 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 $C^L(K)$, that is,
$$u\Pi^L u_{\infty} \leq C^L(K) u_2$$
where $u_2$ denote the $H^2$ seminorm of $u$ and $u_{\infty}$ denote the $L^\infty$ norm of $u$.
The optimal estimation of the constant $C^L(K)$ is expressed as
$$C^L(K) = \sup_{u\in H^2(K)}\frac{u\Pi^L u_{\infty}}{u_2}$$
This project shows the proposed algorithm to obtain the optimal estimate of $C^L(K)$ for any triangle of arbitrary shape. In this algorithm, we consider the solution of the optimization problem involving the maximum norm:
Find $\lambda$ such that
$$\lambda = \inf_{u\in V^L(K)}\frac{(u_2)^2}{(u_{\infty})^2}$$
Note that the objective constant $C^L(K)= \lambda^{1/2}$.
Content Description
The following notebooks are contained in the project.

Maximum Norm Error Estimation
 Here, the framework of the concept and algorithm is explained. The basic concepts and important results are discussed and the concise code for finding the interpolation constant estimate is presented. Given a uniquelydefined triangle using constants $\alpha$, $h_{med}$ and $\theta$, the optimal interpolation constant is estimated using the technique in solving a quadratic optimization problem with maximum norm constraint. In this notebook, the concise function that implements this algorithm is given.\

Comparison of the upper bounds for different mesh size
 In this notebook, for a right isosceles triangle $K$, the estimate for $C^L(K)$ is obtained using uniform triangulation. It is demonstrated how the estimate of the interpolation error constant varies 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. Here, highdegree polynomials are obtained through interpolation at the given nodes of the triangulation. The, since $C^L$ is the supremum of quotients of the form $\frac{ f \Pi^L f _{\infty}}{f_{2}}$, a lower bound for the interpolation error constant is achieved. \

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.\

Interpolation Constants for Various Triangles
 In this notebook, we summarize the estimated interpolation constants, given number of iterations, inner angle, alpha and median length of triangle.\

Interval Computation
 To employ numerical verification, this notebook incorporates interval computation in the algorithm for estimating the interpolation constant. In the first section, interval computation is first verified for a predefined mesh. Next, the concise code for interval computation is presented. A table containing the summary of computations for the interpolation error constant for a right isosceles triangle is also shown.
% Edited Shirley Mae Galindo 08/09/2022
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