The weighted residual method (WRM) by itself is an analytical method that is generally used to obtain an approximate solution. In principal, if you consider a polynomial basis of infinite order, you can always get the exact solution. However, we try to use lower order basis functions to capture the major properties of the solution. In simpler problems this may actually be the same as the exact analytical solution, however, in general, it's mostly an approximation.

This method is also used in numerical solution schemes such as FEM, where the solution domain is discretized into finite-dimensional small elements and then WRM is applied to each element. This allows better resolution with lower order basis functions.