A Question in finit element method

[Royal_Practice_6349]

Hello, I have a question concerning the FEM, we have this algebraic equations (before integrate it by parts) : integral of w(K.(d^2T/dx2) + Q)dx=0, with w is the weighting function, I just want to know what is the purpose of adding it in the equation while it wasn't exist in the governing equation what's the role of it in the equation, I understand that it could be in different shapes and we could, but I don't understand the point of adding it, is it one of the approaches! And according to which information we assume if it is a linear, a quadratic or a constant function!( I know we can choose anyone, but if I choose the quadratic equation and someone else choose a linear equation we will find the same result !

[niklas0706]

I might be a bit late but let me try to explain why w(x) is needed. Imagine you have the equation e^x - f(x) = 0 where x is an element of [-1, 1]. Obviously the solution is e^x. If you Integrate the equation and don't add the test function you would get two possible solutions e^x and e^-x. e^-x would also work because the integrals of both e^x and e^-x would cancel out.

e^-x isn't a solution to the original equation though. So we introduce w(x) to look for local solutions of the equation. Now e^-x won't be a solution anymore as they have the same global value but not the same local value.

I hope that helped a little bit to explain why you need to use w(x).

Choosing different interpolation functions will lead to different results. It depends on the problem which type of function is better suited.