What is the form of local basis functions for 27-nodes cubic element in FEM?

Question

consider the nodes is ,now I want to solve a pde and I'm trying to find local basis of this nodes,my goal is to find a multivariable polynomial with 27 unknown parameters so how can we find it, for the case of 20 nodes I'have found it

Answer 1

If you know the value of $f(x,y,z)$ at these nodes you can approximate the derivative $\partial^j_x \partial_y^k \partial_z^\ell f$ so long as $j,k,\ell\leq 2$. Your polynomial space is thus the polynomials which are of degree $2$ or less in each variable.

The most natural basis for this space is the product of usual Lagrange polynomials in each dimension. For example, if our nodes are $\{0,1,2\}^3$ then the basis element which is $1$ at the origin and $0$ at the other nodes is $$ \phi_{(0,0,0)}(x,y,z) = \frac{(x-1)(x-2)(y-1)(y-2)(z-1)(z-2)}{(0-1)^3(0-2)^3} $$

Edit: this basis is not unique, but it is natural in the sense that it is dual to the "evaluation at nodes" functionals. Brenner and Scott is a standard reference on finite elements. I have also used Ern and Guermond