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# Estimation of the Crouzeix-Raviart constant over triangle element

Given a triangle domain $K$ with vertices $(0,0), (1,0), (\cos \theta, \sin\theta)$, denote the edges of $K$ by $e_1$, $e_2$, $e_3$.

Let $\Pi_h$ be the Crouzeix-Raviart interpolation such that $$ \int_{e_i} \Pi_h u -u ds = 0\quad (i=1,2,3)\:. $$

We consider the estimation of the following constant $C$:

$$ |\Pi_h u -u |_{L^2(K)} \le C | \nabla(\Pi_h u -u) |_{L^2(K)} $$

The constant is evaluated by solving the corresponding eigenvalue problem of Laplace operator.

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