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.

1. 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 uniquely-defined 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.\

2. 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.\

3. Lower Bounds of the Constant

   • In this notebook, the lower bound of the constant is also determined to verify sharpness. Here, high-degree 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. \

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.\

5. 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.\

6. 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