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

Abstract

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

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• An Extended Galerkin analysis in finite element exterior calculus
• An arbitrary-order discrete de Rham complex on polyhedral meshes: Exactness, Poincaré inequalities, and consistency
• Virtual elements for Maxwell's equations
• A polyhedral discrete de Rham numerical scheme for the Yang-Mills equations
• Cohomology of the discrete de Rham complex on domains of general topology
• A discrete three-dimensional divdiv complex on polyhedral meshes with application to a mixed formulation of the biharmonic problem
• Discrete weak duality of hybrid high-order methods for convex minimization problems
• A polytopal discrete de Rham complex on manifolds, with application to the Maxwell equations
• An arbitrary-order fully discrete Stokes complex on general polyhedral meshes
• A polytopal discrete de Rham scheme for the exterior calculus Einstein's equations
• A pressure-robust HHO method for the solution of the incompressible Navier-Stokes equations on general meshes
• An arbitrary-order discrete de Rham complex on polyhedral meshes. Part II: Consistency