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A major challenge which arises when one aims to develop a software package that implements the spectral/hp element method is to implement the mathematical structure of the method in a digestible and coherent matter. Obviously, there are many ways to encapsulate the fundamental concepts related to the spectral/hp element method, depending on e.g. the intended goal of the developer or the chosen programming language. We will (without going in too much detail) give a an overview of how we have chosen to abstract and implement spectral/hp elements in the Nektar++ library. However, we want to emphasise that this is not the only possible choice.

Five different sublibraries, employing this characteristic pattern, are provided in the full Nektar++ library:

This structure can also be related to the formulation of a global spectral/hp element expansion, i.e.

\[ u(\boldsymbol{x})=\overbrace{\sum_{e\in\mathcal{E}}\underbrace{\sum_{n\in\mathcal{N}}\phi^e_n(\boldsymbol{x})\hat{u}^e_n}_{\mbox{\scriptsize{LocalRegions library}}}}^{\mbox{\scriptsize{MultiRegions library}}}=\sum_{e\in\mathcal{E}}\underbrace{\sum_{n\in\mathcal{N}}\phi^{std}_n\overbrace{(\left[\chi^e\right]^{-1}(\boldsymbol{x}))}^{\mbox{\scriptsize{SpatialDomains library}}}\hat{u}^e_n}_{\mbox{\scriptsize{StdRegions library}}}. \]

A more detailed overview of the Nektar++ structure, including an overview of the most important classes per sublibrary, is depicted in the figure below.

Main structure of Nektar++