Galerkin approximation for an elliptic BVP

What is the process for deducing Galerkin approximation for an elliptic boundary value problem? For example, for a problem like--

$$-\Delta u + cu = f\ \ \mathrm{in}\ Ω$$

$$u = g\ \ \mathrm{on}\ \ \Gamma_1$$

$$\frac{∂u}{∂n} + pu = h\ \ \mathrm{on}\ \ \Gamma_2$$

here, $$∂Ω = \Gamma_1 \cup \Gamma_2$$ and $$\Gamma_1\cap \Gamma_2 = \phi$$ and $$c \geq 0.$$

$$f, g, h, p$$ are sufficiently smooth functions.

I’ve calculated the weak solution of this problem to be

$$\int_\Omega (\nabla u\cdot \nabla \phi + cu\phi)dw + \int_{\Gamma_2} (pu-h)\phi ds = \int_{\Omega} (f\phi) dw$$

here $$\Omega, \Gamma_2$$ with the integral sign $$\int$$ are the domain identifier.

So how can I propose a galerkin approximation from this weak form of the PDE

The idea of the Galerkin FEM is to approximate this solution. After the steps you have mentioned, just take a finite element subspace $$V_h\subset V=H^1_0{\Omega}$$. Then, as $$V_h$$ is finite, it will have a finite basis $$\lbrace \phi\rbrace_{i=1}^n$$ where $$n$$ is the dimension of the space. Then you can approximate the solution $$u$$ by $$u_h$$ where $$u_h$$ is a linear combination of $$\lbrace \phi\rbrace_{i=1}^n$$, i.e, $$u_h=\sum_{i=1}^n\phi_iu_i$$. Now using, $$\phi=\phi_i$$ in your equations, you will get a system of equations of the form $$Au=b$$ where $$A$$ is referred to as stiffness matrix and $$b$$ as load vector. You can refer to any standard FEM book for more details.