Nonlinear finite elements/Axial bar approximate solution

Approximate Solution: The Galerkin ApproachEdit

To find the finite element solution, we can either start with the strong form and derive the weak form, or we can start with a weak form derived from a variational principle.

Let us assume that the approximate solution is   and plug it into the ODE. We get

 

where   is the residual. We now try to minimize the residual in a weighted average sense

 

where   is a weighting function. Notice that this equation is similar to equation (5) (see 'Weak form: integral equation') with   in place of the variation  . For the two equations to be equivalent, the weighting function must also be such that  .

Therefore the approximate weak form can be written as

 

In Galerkin's method we assume that the approximate solution can be expressed as

 

In the Bubnov-Galerkin method, the weighting function is chosen to be of the same form as the approximate solution (but with arbitrary coefficients),

 

If we plug the approximate solution and the weighting functions into the approximate weak form, we get

 

This equation can be rewritten as

 

From the above, since   is arbitrary, we have

 

After reorganizing, we get

 

which is a system of   equations that can be solved for the unknown coefficients  . Once we know the  s, we can use them to compute approximate solution. The above equation can be written in matrix form as

 

where

 

and

 

The problem with the general form of the Galerkin method is that the functions   are difficult to determine for complex domains.