## 3 Theoretical Baseline
### Finite Volume Method  
In a similar vein to the FDM, the use of finite volumes might appear distinctly different from the idea of finite elements. Here, we make extensive use of [PERSON] theorem to replace volume with hull integrals in conservation laws [33]. There exist different formulations of this method, most namely, a cell- and vertex-centered form. The main difference lies in where the solution is stored. In the former case, the solution is stored at the polygonal cell centers that are spanned by the mesh vertices. The latter, instead, directly uses these vertices as solution points [34]. In this case, one does not operate on the computational mesh directly but, rather, on its dual. As the cell-centered formulation is considerably more widely used, we investigate this variant further in the following. It can be shown, however, that the FVM can simply be considered a Bubnov discontinuous Galerkin method of polynomial order zero. To illustrate this, we again turn to Equations (1) and (2) describing the strong and weak form of the advection equation. A finite volume approximation in conservation form is  
\[\int_{\Omega}\partial_{t}\alpha\,dx+\int_{\Omega}\underline{u}\
abla\alpha\, dx=\int_{\Omega}\partial_{t}\alpha\,dx+\int_{\partial\Omega}\alpha\,( \underline{u}\cdot\underline{u})\,ds. \tag{16}\]  
Apart from the presence of a test function \(v\) in Equation (2), the second integrand simply represents the net flux of the conserved quantity \(\underline{u}\alpha\) over the set of element boundaries. For Equations (2) and (16) to be equivalent in this case, the third integrand resulting from partial integration has to vanish in addition. However, this can be shown trivially by setting the order of the polynomial space for the trial and test function to zero. Then, the derivative of the test function vanishes and, thus, the entire term does not contribute to the weak form. After performing this step, the test function is still present in the remaining parts of Equation (2). For the remaining terms to be equivalent, they must vanish out of the equation as well. This can be accomplished straightforwardly by fixing the value of the test function to be unity. In the weak form, this step is admissible since it must hold for all instances of \(V\). As \(1\in V^{0}\) where \(V^{0}\) is the space of constant polynomials, this statement holds in particular for a Bubnov Galerkin scheme, as trial and test space must be identical. The qualitative similarity of both schemes is illustrated in Figure 3.  
For both schemes, DoFs are entirely local to the cell and coupling happens through the calculation of a numerical flux--or, in more formal terms, through the evaluation of the hull integral in the corresponding weak form. However, the FVM only stores one DoF per cell, which has notable implications for the calculation of the numerical flux. This means that as a first step, the cell values have to be reconstructed at the mesh facets. These reconstructed DoFs which depend on the cell values that they interpolate between can then be coupled to their counterparts at opposing mesh facets. These relationships are denoted by DoFs being colored identically (coupled) and being transparent (reconstructed) in Figure 3a. For a DGM scheme of first order or higher, this interpolation step is oftentimes not necessary if the quadrature scheme is chosen carefully. For a collocation method (see Section 3.3 for a more thorough discussion), one does not need to tabulate the full list of DoF values at the set of facet quadrature points, but rather, only a small subset of DoFs that are owned by the facet [35]. In summary, the FVM can again be considered as a special instance of the Bubnov type DGM, where the shared polynomial space \(V\) is taken to be of constant order and the test function \(v\) is set to be unity. We note, similarly to the discussion on the FDM, that this simplification of Equation (2) brings with it some computational advantages that can be offset by sacrificing flexibility.