Projection method for integral equations

Summary

A projection method approximates the solution of an operator equation by an element of a finite-dimensional subspace. The main case is an integral equation of the second kind. One chooses subspaces $X_n$ and projections $P_n$ onto them, and asks that the equation holds after applying $P_n$. This gives a finite linear system for the coefficients of the approximation. The Galerkin method (orthogonal projection) and the collocation method (interpolation) are the two main cases.