## 3 Theoretical Baseline
### Finite Difference Method  
At first glance, the finite difference method (FDM) might appear to be conceptually different from the finite element methods given above. Instead of treating the discretized problem in an element-wise manner, the FDM operates on discrete points directly and, per se, lacks a notion of cells in the domain. Yet, both methods still may yield identical results in discretization. A comparison of both approaches is shown in Figure 2.  
In the figures, \(i\) and \(j\) denote vertical and horizontal indices of grid nodes, \(\phi_{i}\) are the FE basis functions, and Roman letters denote indices of cells. Forming, for example, the Laplacian for an FDM requires access to the vicinity of vertex \(i,j\) (red node) in all Cartesian directions (blue-colored nodes). For both methods, gray-marked nodes do not pose a contribution to the value of the central node. For a special case of FEM with quadrilateral elements, the same nodes form contributions to the global basis function \(\phi_{5}\). However, we  
Figure 1: Coupling of global DoFs in the continuous Galerkin (**a**) versus discontinuous Galerkin FEM (**b**), both of first order. In the latter case, DoFs are entirely local to the cell and, thus, receive no contribution from neighboring cells. Weak coupling is only introduced by the additional numerical flux. Coupled DoFs are drawn in identical colors. Figure 2: Comparison of the nodal nature of the FDM (**a**) versus the cell-wise assembly used in the CG FEM (**b**) for an identical, Cartesian triangulation with 9 nodes. Both methods are formulated as first-order approximations. Red-colored nodes signify the points where the PDE is evaluated. Contributions to this node are taken from blue nodes, whereas no contribution—white nodes. now do not evaluate the Laplacian operator directly but, instead, gather contributions from weak form integrals. In the case of \(\phi_{5}\), we have to gather contributions from cells \(I\) to \(IV\). However, one may still show the equivalence of CGM and FDM by investigating the resulting global linear system. We exemplify this claim using the Laplacian as a differential operator and the second-order central stencil. This formulation continues to be widely used as an approximation technique. As will be shown later, this particular choice of operator is especially straightforward to compare with the CG-FEM due to the choice of trial and test functions. On a Cartesian, two-dimensional grid with uniform spacing \(h\) in both directions, the approximation reads  
\[\Delta u\approx\frac{u_{i-1,j}+u_{i,j-1}-4u_{i,j}+u_{i+1,j}+u_{i,j+1}}{h^{2}}. \tag{8}\]  
Such a system in stencil notation will produce a global matrix with main diagonal values 4 and four off-diagonals with entries 1. We now proceed to construct an equivalent CG finite element scheme, where the global system matrix is required to be exactly equivalent to the FD formulation. The weak (CG) Laplacian can be formulated as follows: Find \(u_{h}\in V\) such that for all \(v_{h}\in V\)  
\[\int_{\Omega}v\Delta u\,dx=\underbrace{\int_{\partial\Omega}v(\
abla u\cdot \underline{u})\,ds}_{=0}-\int_{\Omega}\
abla v\
abla u\,dx. \tag{9}\]  
We have in this case introduced the additional restriction that trial and test space be identical, that is, we use a Bubnov Galerkin method. Now, let \(\Omega\) be an identical triangulation to the FD variant using quadrilateral \(\mathbb{Q}^{1}\) elements, that is linear Lagrange elements. Then, the four basis functions spanning the reference element are  
\[\phi_{1}(x,y) =xy-x-y+1; \tag{10}\] \[\phi_{2}(x,y) =x(1-y);\] (11) \[\phi_{3}(x,y) =y(1-x);\] (12) \[\phi_{4}(x,y) =xy. \tag{13}\]  
The finite difference stencil given by Equation (8) only takes into account contributions from nodes that lie strictly horizontally or vertically from the node of interest. As a consequence, the node on the reference quadrilateral that is positioned diagonally from the center node must not have any contribution to the weak form integral, otherwise the resulting linear system cannot be equal.