Nonlinear finite elements/Axial bar approximate solution

Approximate Solution: The Galerkin Approach edit

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






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