Dear All,
I was wondering, why in FE formulation, and in the case that both Dirichlet and Neumann BCS are imposed, the weight function (W or v) on the boundary is considered to be zero?
Would the situation change if the problem is solely Dirichlet BC?
Thank you very much
The requirement of shape functions having compact support can be easily see starting from a simple differential equation, multiplying it for the shape function and integrating over the whole space-time domain. You see after integration by parts that a term vanishes on the boundary and one get this way no longer any regularity requirments on the original variable.
Just have a look https://en.wikipedia.org/wiki/Weak_solution
1. http://www.math.pitt.edu/~dwang/PDE-Evans-review.pdf
2.http://www.rgf.bg.ac.rs/predmet/GO/IV%20semestar/Metode%20matematicke%20fizike/Predavanja/Predavanje%209.pdf
3. https://www.ams.org/journals/bull/1992-27-01/S0273-0979-1992-00266-5/S0273-0979-1992-00266-5.pdf
Your statement is not correct.
You have to distinguish between essential and natural boundary conditions. To see that you have first to multiply your PDE by a test function, this will give you the weak formulation to the problem at hand, second use the Green formula (integration by parts) to get the boundary part on which the boundary conditions of your given problem are to be applied. Sometimes we can have a shape function with 1 on compact [a,b] and zero otherwise as a Dirichlet BC.
I hope this helps.
Thank you all for helpful responses so far.
Dear Mohamed,
Weight function (or test function) belongs to finite element space, while based on my understanding, you are saying that we have to get boundary conditions from strong form and apply it on the weight function. Weight function is a function that we choose. Of course it should be in accordance with mathematics of the problem (in PDE context). B.C.S in strong from are dictated by the physics of the problem. Sorry my discussion is not well elaborated, and despite I believe you are correct I still don't see what make weight function to follow some requirements.
At the end of weak form derivation, we end up with values of weight function at both ends of boundary, which, I think, we should have determined that in our weak statement of the problem.
Dear Amir,
I agree with most of what you just wrote.
Two more precisions, the finite element space are usually functional spaces called Sobolev spaces.
Non homogeneous Dirichlet BC can be transformed into 0 Dirichlet BC problems, it is just a simple change of variable.
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