A discrete de Rham complex with enhanced smoothness

Abstract

Discrete de Rham complexes are fundamental tools in the con- struction of stable elements for some finite element methods. The purpose of this paper is to discuss a new discrete de Rham complex in three space dimensions, where the finite element spaces have extra smoothness com- pared to the standard requirements. The motivation for this construction is to produce discretizations which have uniform stability properties for cer- tain families of singular perturbation problem. In particular, we show how the spaces constructed here lead to discretizations of Stokes type systems which have uniform convergence properties as the Stokes flow approaches a Darcy flow.