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https://cwi-nl-zoom.zoom.us/j/86497148056?pwd=7LgxTgvDMMlgvIWKeyd800bp4Ffgnb.1
Meeting ID: 864 9714 8056 Passcode: 499038
Title: Mantis.jl: FEEC-based structure-preserving discretisations in Julia
Abstract: Structure-preserving discretisations aim to retain, at the discrete level, fundamental invariants and geometric structures of partial differential equations (PDEs). Typical examples include conservation of energy, momentum, and helicity for the Navier–Stokes equations, and potential enstrophy for the shallow-water equations. 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 Finite Element Exterior Calculus (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 — with applications in electromagnetics or diffusion problems — and 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.
Most existing finite element libraries are built around classical vector-calculus formulations and incorporate FEEC concepts indirectly. While effective, this approach often requires ad hoc constructions to recover the underlying geometric structure, and can complicate the generalisation to higher dimensions, non-standard function spaces, or complex geometries.
Mantis.jl is a Julia finite element library designed natively around the FEEC paradigm. It provides a flexible environment for formulating and discretising PDEs directly in the language of exterior calculus, supporting arbitrary spatial dimensions and a wide range of discrete function spaces.
In this talk, we will provide an introduction to the concepts of FEEC and differential forms, and we will show how those are used within Mantis.
