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In the finite element lecture we learned, that it is impractical to use centre of gravity of a triangle as a degree of freedom. So we defined a subspace $P_3'$, where $P_2\subseteq P_3'\subseteq P_3$ $$P_3'=\{p\in P_3;\ p(a_{123})=-\frac{1}{6}\sum_{l=1,2,3}p(a_l)+\frac{1}{4}\sum_{l,m=1,2,3,\ l\neq m}p(a_{llm})\}$$ where $a_1$, $a_2$, $a_3$ form a triangle in 2 dimensions, $a_{llm}=\frac{1}{3}(2a_l+a_m)$.

What is the reason for this? Why don't we use the centre of gravity?

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    It's very possible that you'll get better answers at [Compuational Science Stack Exchange](http://scicomp.stackexchange.com/) to questions like this. I know that there are some smart people who know a lot about FEM over there... – 2017-01-03
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    Thank you. If anyone feels answering there is more appropriate, the question can be found here: http://scicomp.stackexchange.com/questions/25878/why-do-we-eliminate-centre-of-gravity-as-degree-of-freedom-in-fem – 2017-01-03

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