An arbitrary-order fully discrete Stokes complex on general polyhedral meshes

Abstract

In this paper we present an arbitrary-order fully discrete Stokes complex on general polygonal meshes. Based upon the recent construction of the de Rham fully discrete complex [14] we extend it using the same principle. We complete it with other polynomial spaces related to vector calculus operators and to the Koszul complex required to accommodate the increased smoothness of the Stokes complex. This complex is especially well suited for problem involving Jacobian, divergence and curl, like e.g. the Stokes system or magnetohydrodynamics. We show a complete set of results on the novelties of this complex complementing those of [14]: exactness properties, uniform Poincaré inequalities and primal and adjoint consistency. We use our new complex on the Stokes system and validate the expected convergence rates with various numerical tests.

An exterior calculus framework for polytopal methods

“…Despite their non-conformity, polytopal technologies can be used to develop compatible frameworks. Polytopal discretisations of the de Rham complex (1.1) have been proposed, e.g., in [10,33,38], and applied to a variety of models , such as magnetostatics [8,34], the Stokes equations [11], and the Yang-Mills equations [47]; they have also inspired further developments, based on the same principles, for other complexes of interest such as variants of the de Rham complex with increased regularity [32,55], elasticity complexes [19,44], and the Stokes complex [12,14,49]. Polytopal complexes have additionally been used to construct methods that are robust with respect to the variations of physical parameters, in particular for the Stokes problem [11], for the Reissner-Mindlin equation [43], or the Brinkman model [33].…”