## 5 Examples
### Allen-Cahn Equation
developed a custom, high-performance finite element simulation code called MeltPoolDG based on the discontinuous Galerkin method as well as a mixed FEM to resolve the melt pool on the powder bed scale [102; 103]. Other recent developments include a finite element model on multiple grids to capture the governing physics [104], and a space-time FEM code that is based on a Petrov Galerkin approximation [105]. Furthermore, there also exist implementations of such melt pool models using commercially available software. For instance, the popular COMSOL software package (version 5.5) which implements many variants of the FEM can resolve the relevant thermo fluid dynamics during laser melting [106; 107]. ## 6 Conclusions and Future Work

We presented a general framework for determining combinations of interoperable, grid-based approximation schemes that are individually suited for the solution of multi-physics problems. The key findings of this work can be summarized as follows:

* The discontinuous Bubnov Galerkin method can serve as a general mathematical baseline for the finite volume method, finite difference method, and continuous Galerkin method. These form the majority of schemes used in practice, which is why we restrict the scope of this work to this specific subset. We have shown that one can recover all these schemes by introducing certain restrictions to the DG method, which can either be on the algebraic or algorithmic level. * By using those simplified schemes where appropriate, one can gain stability and performance in numerical simulations that scale with problem size and hardware. In particular, one may even avoid the assembly of a global linear system by leveraging domain restrictions in the case of the finite difference method. If a PDE exhibits strong \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Variable** & **D1** & **D2** & **D3** & **D4** & **Result** \\ \hline \(a_{\text{solid}}\) & no & hyperbolic & n.a. & linear & DGM \\ \(\hat{a}_{\text{liquid}}\) & no & hyperbolic & n.a. & linear & DGM \\ \(a_{\text{gas}}\) & no & hyperbolic & n.a. & linear & DGM \\ \(\underline{u}\) & no & parabolic & irregular & n.a. & CGM \\ \(p\) & no & hyperbolic & n.a. & linear & DGM \\ \(T\) & no & parabolic & irregular & n.a. & CGM \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Resulting schemes of the proposed decision method for the laser powder bed fusion example problem. Columns D1 to D4 refer to the corresponding decision points in Figure 5.

nonlinearity, choosing a purely reconstruction-based approach via the finite volume method is then beneficial and assembly of the weak form can be avoided as well. * This common framework based on the DG method enables interoperability of the mentioned schemes. As such, one can combine them when solving multiphysics problems and thus assemble efficient mixed discretizations. Therefore, manual coupling of different numerical solvers is not needed in theory. Instead, one may implement them in a monolithic way and, thus, avoid costly memory transfer operations. * One can systematically derive an efficient combination of numerical schemes, given some inputs about the problem, that is, hardware requirements, the mathematical formulation of the problem, and the domain geometry. The method to identify these schemes is based on the field and delivers one-to-one recommendations on which method to use. * Applying the developed method to the two model problems in this work, the Allen-Cahn equation in 1D and 2D and the advection equation in 2D, yields methods that notably outperform their respective alternatives. In the former case, this difference in computation time exceeds one order of magnitude. In the latter case, we have shown that, as predicted by the theory, the computational advantages scale with problem size and the necessary degrees of freedom for some fixed accuracy differ considerably.