There are several Galerkin methods. One of most known is the Galerkin method of weighted residuals (which is equivalent to the application of the method of perturbations). It converts the differential equation or associated strong formulation to a weak formulation. Also called Galerkin approximation is the interpolation method on the weak formulation, which are used for converting its continuous problems to the respective discrete problems.

In my opinion the main advantage of the Galerkin's method for eigenvalue problems is its ability to provide extremely powerful numerical tools to solve problems with more elaborated geometries, material distributions, etc. that hardly can be treated by other methods.

Other advantages exist ... hope other comments can improve this comment.

 Hi,

Galerkin method is a kind of weighted residual method, where weight functions are same as basis/trial functions.

Galerkin method is also popular in the  finite element method (FEM) since it offers ease of implementation due to same weight and trial functions.

For dynamic mechanical problems the Galerkin method is used as a global/spectral method to help reduce the Partial Differential Equation (PDE) into ordinary differential equations. Vibrational mode shapes are usually used as basis functions (polynomials or some other suitables functions can be used as basis function if mode shapes are not available). Only a few mode shapes will give a converged solution for the problem. It reduces the dimensionalilty of the problem hence it is much faster.

I hope it helps.

Hi,

Implementing Galerkin method in dynamic mechanical problems gives rise to reduce the Partial Differential Equations to Ordinary differential equations (ODE).

W(x,t)=phi(x)*q(t)

where; phi(x) is mode shape