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In 2D, the hat functions are the usual function basis for the finite dimentional space $\{v\in \mathcal{C}:v|_T\in P_1(T)\,\forall T\}$: set of continuously functions that are polynomials less than or equal to one over each element (triangle) $T$ of the mesh.

But, which are the function bases for $\{v:v|_e\in P_k(e)\,\forall e\}$? (space of functions that are polynomials less than or equal to one on edge $e$ of the mesh).

In other words, how can I define a base of polynomials less than or equal to one over an single edge $e$ of the mesh?

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    $\begingroup$ Exactly like you would define the hat functions on a 1D mesh: the two linear functions that are equal to $1$ on one of the two vertices and $0$ on the other. $\endgroup$ Commented Jul 19, 2016 at 7:12

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