A discrete de Rham complex with enhanced smoothness

Abstract

Discrete de Rham complexes are fundamental tools in the construction 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 compared to the standard requirements. The motivation for this construction is to produce discretizations which have uniform stability properties for certain 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. Keywords: Discrete exact sequences, nonconforming finite elements, Darcy-Stokes flow, uniform error estimates. Mathematics Subject Classification (1991): Primary 65N12, 65N15, 65N30. © Springer-Verlag 2006.