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Suppose we are given the followin where $f$,$u$, $g$ are given functions:

$-Delta u = f$ in $Omega$

$u=u_o$ on $Gamma_1$

$frac{du}{dn}=g$ on $Gamma_2$

So in order for me to form the variational problem which is the first step, I did the following.

Assume $u_o$ is sufficiently smooth on $hat{Gamma}$ then, $hat{u}=u-u_o$

This results in:

$-Delta hat{u}= f$ in $hat{Omega}$

$hat{u}=0$ on $hat{Gamma_1}$

$frac{dhat{u}}{dn}=g-frac{du_o}{dn}$

If I let $V$={$v in H^1(Omega): v|_{Gamma_1} = 0$}

Then by applying my test function v and using Green's Formula, I can show that the new problem I've created is similar to the original problem and it results in:

$int_{Omega} nabla u nabla vdx$ = $int_{Omega}fvdx+int_{Gamma_2}gv ds $

where the bilinear case is $a<u,v> = int_{Omega} nabla u nabla vdx$

the linear case is $L(v) = int_{Omega}fvdx+int_{Gamma_2}gv ds $

So now that I have gotten the variational problem finished I need to implement some finite element method.

So this would mean I need to find a basis function which I would pick a triangulation in some smaller subspace $V_h=${$v_h$ is continous, linear.},

$phi_j(x)= 1$ if $i=j$ and $0$ if $ineq j$

So then,

$v_h=sum_{j=1}^{M} eta_j phi_j(x)$ are the linear combinations

Now essentially I need to get to some sort of formulation of the finite element method where I would have

$A zeta=b$ and then I could get some sort formulation of what the A matrix would look like. I know the A matrix will be symmetric and positive definite.
I know the following,

$A_{ij}= <phi_i' , phi_j'>$

$b=<f, phi_i>$

I'm stuck on the finite method part and I want to make sure I'm going about it the right way.