I am trying to solve this problem:
so far I could do some math to reach here:
Multiply equation (1) by a test function $v$ and integrate over the domain $\Omega$. Then, we apply integration by parts to get the weak form of the problem:
Find $u$ such that $u = 0$ on $\Gamma_1$ and
\begin{equation} \int_{\Omega} \nabla u \cdot \nabla v \, dx = \int_{\Omega} 2v \, dx + \int_{\Gamma_2} v \, dS, \quad \text{ with } v = 0 \text{ on } \Gamma_1. \end{equation}
The bilinear form $a(u, v)$ and the linear functional $F(v)$ are:
\begin{align*} a(u, v) &= \int_{\Omega} \nabla u \cdot \nabla v \, dx, \\ F(v) &= \int_{\Omega} 2v \, dx + \int_{\Gamma_2} v \, dS. \end{align*}
My question is what exactly should i do next and how to proceed? What should be v? how to impliment it in the mesh? how to calculate the gradients etc and finally get to the solution u? Can someone help me step by step? Thanks.