Variational Methods: The Ritz Method

The Ritz Method is a precursor to the finite element method. Here is a code that provides a solution to an ODE whose solution is known.

The Ritz Method

Problem: Consider the differential equation

-d^2u/dx^2 - u + x^2 = 0 for 0<x<1;

with the boundary conditions:

u(o) = 0, u(1) = 0.

The Ritz method yield an N-parameter Ritz solution U_N = C_1phi_1 + C_2phi_2 + ... C_iphi_i + ... C_Nphi_N, where phi_i = x^i - x^(i+1)

and C_is are obtained from

K_ijC_i = F_i. In this case, K_ij = B(phi_i, phi_j) = int_0^1(dphi_i/dx *dphi_j/dx - phi_iphi_j)dx is the bilinear part of the variational problem and and F_i = -int_0^1(x^2phi_i)dx is the linear part.