## 4 Method for Assembling Numerical Schemes
### Preliminary Assumptions  
As a starting point, it has to be stated that encompassing the entire state of research on such schemes would be an impossible task. The likewise formalization of a common framework is equally challenging as a consequence and, thus, is not considered in this work. Instead, we follow the path of introducing some restrictions that are, on one hand, enough to construct a unifying scheme but, on the other hand, not too strict such that the efficient solution of real-world problems would be out of scope. Thus, we propose the following restrictions to arrive at a one-to-one choice of numerical schemes. These apply to single-field, as well as to multiphysics problems. 1. Only Bubnov Galerkin schemes are considered, that is, we omit Petrov Galerkin methods. The former restricts the choice of test space to be identical to the trial space. As such, we omit schemes that, for instance, use weighted functions or stencils to account for flow fields. An example of such schemes would be the streamline \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Scheme** & **Geometry** & **Function Space** & **Weak Form** & **Quadature** \\ \hline DGM & Arbitrary & Discontinuous (\(L^{2}\)) & Full & Arbitrary \\ CGM & Arbitrary & Continuous (\(H^{1}\)) & No hull integrals & Arbitrary \\  & & & over interior facets & \\ FDM & Cartesian & Continuous (\(H^{1}\)) & No hull integrals & Collocation \\  & & & over interior facets & \\ FVM & Arbitrary & Discontinuous (\(\mathbb{P}^{0}\in L^{2}\)), & No volume integrals & None required \\  & & Bubnov Galerkin & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of the individual restrictions that the presented schemes impose. Certain simplifications bring with them computational advantages, as discussed above. upwind Petrov Galerkin (SUPG) method [32]. This restriction is essential to obtain an unambiguous choice of method, as the notion of Petrov Galerkin methods does not imply any particular choice of function space. Naturally, omitting the use of arbitrary test function spaces contains a trade-off as we, in theory, restrict the solution space by doing so. However, the majority of numerical methods developed so far indeed fall within the category of Bubnov Galerkin schemes. Thus, we mainly disregard special instances of numerical methods that are tailored to particular use cases, such as the SUPG method outlined above. 2. We omit function spaces for approximation other than the \(L^{2}\) and \(H^{1}\) Sobolev spaces. There exist a vast variety of so-called mixed finite element schemes that use finite elements based on different or composite function spaces with unique properties [42]. For example, one may construct function spaces that can exactly fulfill divergence-free properties (\(H(\operatorname{div})\)) or conditions based on the rotation of a field (\(H(\operatorname{curl})\)). The specific choice of finite element, then, would require a considerable amount of expertise and would warrant a complex decision process of its own. Furthermore, the number of elements available for such spaces is vastly ambiguous, as outlined in Section 1. We thus focus on scalar-, vector-, and tensor-valued Lagrange elements solely. They have been shown to encompass a similar solution space as well and perform comparably for fluid and electromagnetic problems [43; 44]. In summary, we similarly sacrifice some freedom in choosing possibly very-well-suited function spaces for the benefit of attaining a decision metric that allows for an unambiguous recommendation. 3. Closely related to the previous statement, we restrict the solution space further by requiring that only finite elements utilizing Lagrange polynomials should be used. As the standard scalar- and vector-valued \(\mathbb{P}^{k}\) and \(\mathbb{Q}^{k}\) finite elements, being by far the most popular choices, use exactly this family of polynomials, this requirement is weaker in practice than it might seem at first glance [45; 46]. 4.