Consider the problem $$ -u'' = f \ \ \text{in} \ (0,1), \\ u(0) = (1) = 0. $$ Assume that the Green's functions of the nodal values $G(x_j, \cdot)$ lie in $V_h = \{ v \in C([0,1]) : v \ \text{is linear on each interval } [x_{j-1}, x_j] \}$. I need to show, that the
FEM solution $u_h$ is identical to the interpolant $I_h u$ of the exact solution.
I can show that the finite difference method give the same result as the finite element method and then show, that FDM solution is identical to the interpolant $I_h$.
How can I show it using Green's function?