# I Finite Element Method: Weak form to Algebraic Equations?

1. Oct 8, 2016

### ramzerimar

Okay, I'm following a series of video lectures on applications of finite element method to engineering, and the tutor started by demonstrating the mathematical background of FEM using a simple heat transfer problem. He derived the governing equation (in just one dimension):

(1) $$k\frac{d^2 T}{dx^2} + Q = 0$$

Where K is a constant, T is temperature and Q is the heat generated. The next step was discretizing the domain (in this case, a bar with lenght L). For this, we used the weighed integral form, which is:

(2) $$\int_{0}^{L}w_e(k\frac{d^2 T}{dx^2} + Q) = 0$$

Where w is a arbitrary linear weighting function. I understand that we can't solve (1) by using this discretization, because the temperatures would be discontinuous at the nodes and the second derivative wouldn't be defined, and that's why we integrate it to get only first derivative terms. But I didn't understand the weighting term. Just integrating it wouldn't be enough? What's the purpose of it?

Last edited: Oct 8, 2016
2. Oct 8, 2016

### Paul Colby

I think the author is backing his way into a variational expression for the equation of motion. The equation written is valid for all reasonable choices of the test function. This allows for a simple matrix form of the equation being modeled that is easy to program.