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In this work, merging ideas from compatible discretisations and polyhedralmethods, we construct novel fully discrete polynomial de Rham sequences ofarbitrary degree on polygons and polyhedra. The spaces and operators thatappear in these sequences are directly amenable to computer implementation.Besides proving exactness, we show that the usual three-dimensional sequence oftrimmed Finite Element spaces forms, through appropriate interpolationoperators, a commutative diagram with our sequence, which ensures suitableapproximation properties. A discussion on reconstructions of potentials anddiscrete $L^2$-products completes the exposition.

Fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra