Lagrange Interpolation Method:

In FEM, Lagrange interpolation method is used for the polynomial interpolation.

The formula was named after Joseph Louis Lagrange who published it in 1795, though it was first published by Edward Waring in 1779 and rediscovered by Leonhard Euler.

Polynomial U(x) of degree ≤ n – 1 passes through n points (x_1,u₁ = f(x₁)), (x_2,u₂ = f(x₂)),….., (x_n,u_n = f(x_n)) is given by :

U(x) = ∑ U_j (x) where j = 1 to n.

Or U(x) can be written explicitly as,

U(x) =