Revisiting the commonalities of FEEC, Finite Volumes, and Finite Differences
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Structure-preserving discretisations aim to retain, at the discrete level, fundamental invariants and geometric structures of PDEs. Typical examples include conservation of energy, momentum, and helicity for the Navier–Stokes equations, potential enstrophy for the shallow-water equations, and magnetic and cross helicity in magnetohydrodynamics. Beyond their conceptual relevance, such discretisations often lead to concrete numerical advantages, including enhanced stability, the elimination of spurious modes, and improved long-time accuracy. The FEEC framework provides a rigorous mathematical foundation for the construction of structure-preserving finite element discretisations. FEEC is based on the identification and discretisation of Hilbert complexes underlying the continuous problem, such as the de Rham complex in electromagnetics, the Stokes complex in incompressible flow, or the elasticity complex. Its formulation relies heavily on differential-geometric concepts: physical fields are represented as differential forms, differential operators are unified through the exterior derivative, and constitutive and metric relations are encoded via Hodge-* operators. Extensions of this framework, including vector-valued differential forms, are increasingly relevant with particular emphasis in the context of multiphysics applications. Substantial work on structure-preserving discretisations has been carried out in the contexts of finite difference and finite volume methods. However, relatively little effort has been devoted to establishing systematic connections with finite element methods, and FEEC in particular. Much of this work has developed independently, and the common principles, shared structures, and unifying concepts underlying structure-preserving discretisations across different numerical approaches are rarely made explicit. Building on Matussi's work, this contribution discusses the intrinsic similarities between these three canonical discretisation techniques from a structure-preserving perspective. In particular, we emphasise the discrete nature of physical field laws and their canonical discretisation, which is common to all approaches. We further analyse the distinctive role of constitutive relations (material properties), showing how they encode the numerical approximation and the method-specific features of each discretisation technique, and how they correspond to different choices for interpolating (reconstructing) the discrete solution.
