Galerkin method

Summary

The Galerkin method approximates the solution of a problem in weak form, $B(u, v) = f(v)$ for all $v$ in a Hilbert space. It solves the same problem on a finite-dimensional subspace. This turns a partial differential equation into a finite system of linear equations. For a bounded and coercive form, the Lax–Milgram theorem gives existence and uniqueness of the approximation. Its error is within a fixed factor of the best possible approximation from the subspace. The finite element method and spectral methods are Galerkin methods, and Galerkin approximations are also used to prove existence of weak solutions.