### Title

Interpolation error estimate

### Description

Demonstration of the method of estimating the upper bound of the constant for the maximum norm of the Lagrange interpolation error.

### Properties

Public

• DIRECTORY
• DEMO

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.

1. Maximum Norm Error Estimation

• Here, the framework of the concept and algorithm is explained.
2. 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.
3. Lower Bounds of the Constant

• In this notebook, the lower bound of the constant is also determined to verify sharpness.
4. 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

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