A DIALOGUE ON WEIGHTED RESIDUAL METHOD

Student: What is Galerkin formulation?
Dr Airil: Galerkin formulation is a type of weighted residual method aka WRM.

Student: What is a WRM then?
Dr Airil: WRM is one of the numerical methods for solving DE.

Student: How WRM does that (solving the DE)?
Dr Airil: Ok, lets say we have a DE of a domain. To make thing simple, lets take this domain as 1-dimensional, having a length of L. Then assume a trial function and insert this trial function into the DE. Since this trial function is not the actual solution of the DE (if it is the actual solution, we would not have the problem in the first place right because we already know the solution), there will be a so called residual function, R which in the forms of ordinary function rather than DE. Since it is now an ordinary function, it can be integrated over the domain, that is from 0 to L. Such integration will 'replace' the independent variable say, x into numeral and leave the coefficient of the trial functions as the unknowns. Then we will end up with algebraic equation instead of DE.

Student: But wait, you have talked about this, haven't you?
Dr Airil: Yes I did when we were discussing about the basic idea of any numerical techniques.

Student: Ok, please proceed.
Dr Airil. Thanks, I will continue. But, if we conduct the integration on R only once, we will ended up only with one equation which is not enough because we have many unknown coefficients. So we need more equations, as many as there are the coefficients. Say, if we have three coefficient, that means we need three algebraic simultaneous equations. So, the question is how can we get this three equations? Or the more general question would be, if we have n unknown coefficient, how can we get n simultaneous algebraic equations? Can you follow me?

Student: Yes, I can follow you so far. Please continue.
Dr Airil: Ok. The answer is, we must create the n equations. And this can be done by multiplying R with another function say w before we do the integration. Because every time we multiply R with a new function,w which is independent of R, we can get a totally 'new' function of Rw. So, if we need to create say three algebraic equations, all we need to do is to create functions Rw1, Rw2 and Rw3. But there is one last condition, w1, w2 and w3 must all be independent from each other. After that, we conduct the individual integration on these three 'new' functions to obtain the desired set of simultaneous equations. That's it, this is WRM.

Student: If this is WRM, then what is Galerkin?
Dr Airil: Galerkin is when we employ shape functions as the interpolation functions but since we have been talking for quite some time now, I think its better for me to postpone any further discussion on Galerkin although that was what you initially asked about. I want you to digest first what I have told you about WRM above.

Student: Ok understood. But before I let you go, I like you to know that what you have told sounds a bit difficult to me.
Dr Airil: Do not worry. Please take this discussion as a preparation for tomorrow's night FE gathering. I can assure you that things will be much easier once you see me 'in action', you know what I mean.

Student: Oh, really? If that is the case, I can't wait to see you tomorrow.
Dr Airil: I am glad to hear that. Tomorrow then.