# Space-Time FEM for parabolic problems

I am trying to solve a parabolic problem (an IBVP) in one spatial variable using the Galerkin method. After searching for inspiration, I find that the typical approach is to discretise the temporal domain using finite differences and applying FEM in space. However, I would like to solve my problem using FEM in both dimensions. The definition of a weak solution is a function $u$ satisfying:

$u \in L^2(0,T;H_0^1(\Omega))$, $\quad u'\in L^2(0,T;H^{-1}(\Omega))$:

$(i)\quad \langle u',v \rangle + B(u,v;t) = (f,v)\qquad \forall v\in H_0^1(\Omega)\text{ a.e.} \;\; t\in [0,T]$

$(ii)\quad u(0) = g$

where $\Omega = [0,K]$ for some constant $K$ in my case.

It is not obvious to me how to use this in order to arrive at a problem that can eventually be solved on a mesh. I have handled elliptic problems with ease where the transformation is obvious (i.e. finite-dimensional projection, introduce basis functions based on the triangular elements and barycentric coordinates, derive linear system, etc.), but I have not been successful in applying the same method for the parabolic problem. I have been told that this should be possible, but I do not see how given the space in which we search for $u$. Can you point me in the right direction?