## 4 Method for Assembling Numerical Schemes
### Problem Scale  
Another deciding factor for whether adaptivity is needed or not is the presence of multiple length scales in a multiphysics model. We follow the definition given in [56, 57] and characterize a PDE-governed problem to have a Multiscale nature if models of multiple spatial or temporal scales are used to describe a system. Oftentimes, this is the case if equations are used that originate from different branches of physics, such as continuum mechanics versus quantum mechanics or statistical thermodynamics. This may, on one hand, be a physical process with slow and fast dynamics, for example, in chemical reaction networks. Then, the multiscale nature shows itself in the time domain of the problem. Another example of a commonly encountered problem in alloy design is the evolution of the temperature field and phase kinetics during heating and solidification. In this case, various length scales can be involved, such as in processes involving laser heating. The temperature gradients then involve resolutions at a scale of around \(1\times 10^{-5}\) m, whereas the width of a solidification front, rather, reduces to a submicrometer scale, that is, around \(1\times 10^{-7}\) m [58]. Regarding the previous definition, we have one model that is governed by laws of macroscale thermodynamics (that is, the heat equation). The other part of physics present is typically described by the evolution of a phase field. The corresponding equations of this model are, however, derived from the formulation of a free energy functional from Landau theory [59]. Due to the wide variety of physical processes and combinations thereof, formulating general criteria for the presence of a multiscale problem from a mathematical point of view is challenging. To the knowledge of the authors, there do not exist any metrics in the literature that would enable such a classification. We instead rely on the knowledge of the application expert who we assume to be familiar with the physics that should be captured. For a rough disambiguation, however, one may use the definition given above. Such multiscale phenomena are prohibitively expensive to resolve on a uniform mesh due to the nonnegligible difference in the dynamics of the system. One option to efficiently resolve the physics at multiple scales is to employ different grids and solve the resulting problem in parallel. This has, for example, been conducted for the abovementioned case, specifically metal additive manufacturing [60]. A rather effective, alternative approach is the modification of the governing equations such that they become tailored to a specific numerical scheme. For instance, the well-known phase field model has been adapted using specialized stencils to the FDM such that spurious grid friction effects are eliminated [61, 62]. This approach, however, requires extensive knowledge about the numerics as well as the physical nature of a given problem. Another possibility that requires fewer adaptions of the code to the specific problem is to make use of grid adaptive algorithms. This approach for the problem presented is a popular alternative and has been implemented multiple times [63, 64, 65, 66]. Thus, grid adaptivity plays a key role in creating solutions to such problems, if the domains are not to be resolved on different discretizations entirely. Numerical methods as a consequence need to reflect on this requirement; as such, finite difference methods are not suitable for such types of problems. CG finite element methods do enable grid as well as polynomial degree adaptivity. Yet, the imposition of hanging node constraints is oftentimes not trivial. Though there have been considerable strides toward easy and intuitive handling of hanging nodes for continuous elements [67, 68], these methods naturally fall short of the inherently decoupled nature of DoFs present in discontinuous methods.