# Interpolation error constant estimation

## 1. Introduction

This page provides online computation for several interpolation error constants related to $P_0$, $P_1$ and $P_2$ interpolation over specified triangle element $T$. For example, for the Lagrange interpolation $\Pi_h$ that interpolates $u\in H^2(T)$ by polynomial of degree one, the error estimation is given as follows $|u-\Pi_hu |_{H^1(T)} \le C |u|_{H^2(T)}.$ The optimal constant $C$ in the above estimation can be characterized by $C = \sup_{u\in H^2(T)} \frac{|u-\Pi_h u|_{H^1(T)}}{ |u|_{H^2(T)} }\:.$ Such a constant can be evaluated by solving an eigenvalue problem. The computation results in this page will give the exact upper bounds for the constants.

## 2. Parameterization of $T$ and notation

• The nodes of $T$: $p_1$, $p_2$, $p_3$.
• The edges: $e_1$, $e_2$, $e_3$.
• The mid-points of edges: $p_{12}$, $p_{23}$, $p_{31}$.
• Polynomial of degree less or equal to $k$ over $T$: $P_k(T)$.

## 3. Various interpolation operators and error constants:

### Notation

• The $i$th kind interpolation of degree $k$: $\Pi_k^{(i)}$.
For interpolation operator $\Pi_k$ of degree $k$, i.e., $\Pi_k u \in P_k(T)$, if there are more than two kinds of definition, let us distinguish the interpolations with index $i$, i.e., $\Pi_k^{(i)}$.
• Interpolation error constant: $C_{k}^{(i)}$, $C_{k}^{(i,j)}$.
For interpolation operator $\Pi_k^{(i)}$, the interpolation constant is denoted by $C_{k}^{(i)}$. In case that there are more than one type of error constants, we further introduce index $j$ to count the number of constants. That is, $C_{k}^{(i,j)}$.

### List of constants:

• Degree 0: $C_0^{(1)}, C_0^{(2)}$
• Degree 1: $C_1^{(i,0)}, C_1^{(i,1)}, \quad (i=1,2,3)$
• Degree 2: $C_2^{(i,0)}, C_2^{(i,1)}, \quad (i=1,2,3)$

### Interpolation function of order $0$:       $\Pi_0 u \in P_0(T)$

 Interpolation Definition Error estimation $\Pi_0^{(1)}$ $\Pi_0^{(1)}u ={ \int_{T}u d {x} d {y}}/{|T|}$ $\|u-\Pi_0^{(1)}u\|_0 \le C_0^{(1)} |u|_{1}$ $\Pi_0^{(2)}$ $\Pi_0^{(2)}u ={ \int_{e_1}u d {s}}/{|e_1|}$ $\|u-\Pi_0^{(2)}u\|_0 \le C_0^{(2)} |u|_{1}$

### Interpolation function of order $1$:      $\Pi_1 u \in P_1(T)$

 Interpolation Definition Error estimation $\Pi_1^{(1)}$ $(\Pi_1^{(1)}u - u) (p_i)=0 \:\ (i=1,2,3)$ $\|u-\Pi_1^{(1)}u\|_0 \le C_1^{(1,0)} |u|_{2}$,    $|u-\Pi_1^{(1)}u|_1 \le C_1^{(1,1)} |u|_{2}$ $\Pi_1^{(2)}$ $\int_{e_i} \Pi_1^{(2)} u ds= \int_{e_i} u\:ds\:\: (i=1,2,3)$ $\|u-\Pi_1^{(2)}u\|_0 \le C_1^{(2,0)} |u|_{2}$,    $|u-\Pi_1^{(2)}u|_1 \le C_1^{(2,1)} |u|_{2}$ $\Pi_1^{(3)}$ $\Pi_1^{(3)}u (p_i)=u(p_i) (i=\{12\},\{23\},\{31\})$ $\|u-\Pi_1^{(3)}u\|_0 \le C_1^{(3,0)} |u|_{2}$,    $|u-\Pi_1^{(3)}u|_1 \le C_1^{(3,1)} |u|_{2}$

### Interpolation function of order $2$:       $\Pi_2 u \in P_2(T)$

 Interpolation Definition Error estimation $\Pi_2^{(1)}$ $(\Pi_2^{(1)}u -u )(p_i)=0$   $(i=1,2,3,\{12\},\{23\},\{31\})$ $\|u-\Pi_2^{(1)}u\|_0 \le C_2^{(1,0)} |u|_{2}$,   $|u-\Pi_2^{(1)}u|_1 \le C_2^{(1,1)} |u|_{2}$ $\Pi_2^{(2)}$ $(\Pi_2^{(2)}u -u )(p_i)=0$ $(i=1,2,3)$     $\int_{e_i}\Pi_2^{(2)}u -u ds = 0\: (i=1,2,3)$ $\|u-\Pi_2^{(2)}u\|_0 \le C_2^{(2,0)} |u|_{2}$,   $|u-\Pi_2^{(2)}u|_1 \le C_2^{(2,1)} |u|_{2}$ $\Pi_2^{(3)}$ $(\Pi_2^{(3)}u -u )(p_i)=0$ $(i=1,2,3)$     $\int_{e_i} \nabla (\Pi_2^{(3)} u-u)\cdot \! n ds = 0 \: (i=1,2,3)$ $\|u-\Pi_2^{(3)}u\|_0 \le C_2^{(3,0)} |u|_{2}$,   $|u-\Pi_2^{(3)}u|_1 \le C_2^{(3,1)} |u|_{2}$

## 5. Online constant estimation

 Interpolation degree 0 1 2 Interpolation type: Triangle integration Edge integration Triangle vertices:(Input x,y coordinates) p1:   p2:   p3:   Edges: $e_1:=p_1p_2$. Mesh size: (Range: 0.01 ~ 0.1) (The evaluation of constants is based on the finite element method, which needs a mesh for the triangle element $T$. ) Constant values:

## 7. Referece

• Foundament theroem about eigenvalue esitmation:
• Xuefeng Liu, A framework of verified eigenvalue bounds for self-adjoint differential operators, Applied Mathematics and Computation, 267, pp.341–355, 2015 (link, PDF)
• Eigenvalue bound for the Laplacian:
• Xuefeng Liu and Shin'ichi Oishi, Verified eigenvalue evaluation for Laplacian over polygonal domain of arbitrary shape, SIAM J. Numer. Anal., 51(3), 1634-1654. (link, PDF)
• Introduction of various interpolation constants:
• Xuefeng Liu and Fumio Kikuchi, Analysis and estimation of error constants for P0 and P1 interpolations over triangular finite elements, J. Math. Sci. Univ. Tokyo, 17, p.27-78, 2010 (PDF)

Created by Xuefeng LIU, Sep 2nd, 2014.
Last updated: March 30, 2017