I want to solve this by Finite Element Method numerically, since the exact solution is too hard.
Separation of variables does not help me here.
Epsilon is positive so cannot be Helmholtz equation. It seems to be some sort of reaction-diffusion equation. So here should be done more partial differential equations after the strong form has been changed into the weak for form by multiplying both sides by v.
I think that you can assume
$u'(1) = 0$ and $v(0) = 0$.
How can you find the numerical value of this equation?
You should take a look at the FEniCS library!
http://fenicsproject.org/documentation/tutorial/index.html
http://fenicsproject.org/documentation/tutorial/more.html
https://launchpadlibrarian.net/83776282/fenics-book-2011-10-27-final.pdf
It is freely available, and allows you to solve PDEs using finite element methods in a highly automated and easy to use manner. There is a Python interface to it (see the tutorial).