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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) [tex] k\frac{d^2 T}{dx^2} + Q = 0 [/tex]

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) [tex]\int_{0}^{L}w_e(k\frac{d^2 T}{dx^2} + Q) = 0[/tex]

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?

(1) [tex] k\frac{d^2 T}{dx^2} + Q = 0 [/tex]

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) [tex]\int_{0}^{L}w_e(k\frac{d^2 T}{dx^2} + Q) = 0[/tex]

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?

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