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

3. Various interpolation operators and error constants:


List of constants:

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

InterpolationDefinitionError 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 $1$:      $\Pi_1 u \in P_1(T)$

InterpolationDefinitionError 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 $2$:       $\Pi_2 u \in P_2(T)$

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

4. Preparation for online computing

5. Online constant estimation

Interpolation degree
Interpolation type:
Triangle vertices:
(Input x,y coordinates)
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

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