The bubble function In the finite element method and more precisely the MINI element method in two dimensions, they use a function called the "bubble function" which is related to a triangle K of the space meshing and is defined using the following basis functions related as well to the triangle K: $$ \begin{align*} \phi_1(X)&=1-x-y\\\ \phi_2(X)&=x\\\ \phi_3(X)&=y\\\ \end{align*} $$ Where $X=(x,y)$ is a point of the boundary of $K$. Now in all articles and books I consult, they define the bubble function as following: $$ \phi_b(X)=27\phi_1(X)\phi_2(X)\phi_3(X) $$ My problem is I dont get where the 27 came from, I tried to use interpolation functions but I'm not sure if the term 27 comes actually from an interpolation !! Does any one have a clue or maybe the answer ? Thank you for your time.

Essentially, you are considering the product $a·b·c$ for positive $a,b,c$ with $a+b+c=1$. The range of that expression can be bounded above via the inequality of the arithmetic and geometric mean $$ \sqrt[3]{a·b·c}\le\frac{a+b+c}3=\frac13 $$ and this bound is met for $a=b=c=\frac13$. Thus the range is $[0,\frac1{27}]$.

Or in other terms, $27·a·b·c$ has the range $[0,1]$.