In the finite element method we consider a grid with nodes $x_i$, $i=1,...,n$ and define basis functions $\psi_i$ such that $$\psi_i(x_i)=1$$ $$\psi_i(x_j)=0 \text{ for } j\neq i$$ and $\psi_i$ is linear in each finite element $\tau$.
- Why do we say that $\{\psi_i\}_{i=1}^n$ is a Lagrangian basis?
Any function $v$ in the finite element space takes the form $$v=\Sigma_{i=1}^n v_i ~\psi_i.$$ Then the book I'm reading states that since $\psi_i$ is a Lagrangian basis, we see that $v_i=v(x_i)$ and thus, there is a one-to-one mapping between $v$ and the coefficient vector $(v_i)^n_{i=1}$.
- But $v(x_i)=\Sigma_{i=1}^n v_i ~\underbrace{\psi_i(x_i)}_{=1}=\Sigma_{i=1}^n v_i$. It would make more sense to consider $v_j=v(x_j)$ for any but fixed $j \in \{1,...,n\}$.
Is there any reason why it's considered the variable $i$ in the book?
If you want to use $i$ outside the sum, then change the name of the running variable bound to the sum to something else. In computer programming you get the same problem, that variables in local scopes overwrite more global variables with the same name, for the duration of the scope.
"Lagrangian" is exactly the property $ψ_i(x_j)=\delta_{i,j}$ that then makes the coefficients $v_i$ the function values at $x_i$.
So 1. is just giving a name to a set of equations and 2. is a scoping problem.