# boundary conditions in finite element method

In the appendix A of this paper, https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.227.5073&rep=rep1&type=pdf, a finite element method is demonstrated to price a straddle. The same method can be used for an up and out call barrier option.

My question is regarding the boundary condition. In (A.6), the boundary conditions vector involves the quantities $$u_b^n - u_b^{n-1}$$, with $$u_b^n$$ the option price at the time step $$n$$ and the barrier $$b$$. For an up and out call option, wouldn't $$u_b^n - u_b^{n-1} = 0$$ for every $$n$$ ? Or in general, wouldn't $$u_b^n - u_b^{n-1} = 0$$ for all $$n$$ since the time is not relevant as long as the option is at the boundary?

Thanks