Piecewise Polynomials in FEM

Question

In FEM, why is it required to have the piecewise polynomials, $\phi_i(x_j) = \delta_{ij}$ i.e. what is the significance of Kronecker Delta?

Answer 1

In FEM polynomial functions are usually used. As an example of a finite element let us consider a linear tringle with corner nodes $P_1,P_2,P_3$. Polynomials on this element are of the form $a+bx+cy$, i.e. linear combinations of functions $1,x,y$. It is much more convenient to describe arbitrary polynomials on this element by values at corner nodes $P_1,P_2,P_3$ and not by coefficients $a,b,c$. Why? If you find three polynomials, which satisfy \begin{align} \phi_i(P_j)=\delta_{ij}~~~~~~i,j=1,2,3 \end{align} where \begin{align} \delta_{ij}=\begin{array}{l} 1~~\text{for}~i=j\\ 0~~\text{otherwise} \end{array} \end{align} i.e., which equal to $1$ at one node and equal to $0$ at all other nodes, then an arbitrary polynomial with values $A_1$ at node $P_1$, $A_2$ at node $P_2$ and $A_3$ at node $P_3$ can be directly written as: \begin{align} A_1\phi_1+A_2\phi_2+A_3\phi_3 \end{align} And this is a convenient form. However note, that the choice to determine polynomial by values at different nodes is only one option, in Hermitian elements for example, derivatives at nodes are also used.