I am trying to determine the global system of equations of a weak form without specified boundary conditions. Consider, as example, the following weak form:
$$\int_0^2[u'(x)v'(x)+u(x)v(x)]\mathrm{d}x=10v(1)+u'(2)v(2)-u'(0)v(0)$$
where $v$ is the test function, $u$ is the solution function. The term $u'(2)v(2)-u'(0)v(0)$ results from an integration by parts and $10v(1)$ results from $10\int_0^2 \delta(x-1)v(x)\mathrm{d}x=10v(1)$.
My question is how should I deal with the term $10v(1)+u'(2)v(2)-u'(0)v(0)$? Should I use them on the development of the local system of equations? Should I use them on the global system of equations? And how could I do it?
[Note: I intend to solve this problem for any kind of boundary conditions. I know if the boundary conditions are of the Dirichlet type, $u'(2)v(2)-u'(0)v(0)=0$ and the problem get simpler. Anyway, I still wouldn't know how to determine $v(1)$ on the term $10v(1)$.]