A discrete de Rham complex with enhanced smoothness
By Xue-Cheng Tai, Ragnar Winther. 2006
98 citations recorded by the source corpus.
AbstractDiscrete 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
Subfields: Computational Mechanics, Computational Theory and Mathematics, Numerical Analysis
View Paper: https://www.semanticscholar.org/paper/d217e32d57c43eb984183f61a332ab14288f5b25
Citations in the corpus
- A family of nonconforming elements for the Brinkman problem (2012, unknown): “Again, to make this last statement more precise we state a theorem that is similar to a result stated in [37].”
- UNIFORMLY-STABLE FINITE ELEMENT METHODS FOR DARCY-STOKES-BRINKMAN MODELS (2008, unknown): “18) The remaining proof is essentially from Lemma 3 by Tai and Winther [39].”
- Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book (2012, unknown): “, 2002), Tai-Winther (Tai and Winther, 2006), and Bell (Ciarlet, 2002) element families.”
- Robust Globally Divergence-Free Weak Galerkin Methods for Stokes Equations (2016, unknown): “For stable and locally divergence-free Stokes elements of nonconforming types, we refer to [32] for the Crouzeix-Raviart element method with piecewise constant approximation for the pressure, and to [50, 60, 67] for finite element approximations based on modifying H(div)-conforming elements on gener”
- On the Divergence Constraint in Mixed Finite Element Methods for Incompressible Flows (2015, unknown): “Another class of H(div,Ω)-conforming methods for the Stokes problem modifies the Raviart– Thomas and BDM spaces locally with divergence-free vector fields such that the resulting spaces possess weak tangential continuity [32, 47, 61, 66].”
- Preconditioning discretizations of systems of partial differential equations (2011, unknown): “Nonconforming finite elements, which are uniformly stable, have been constructed in [94] in the two-dimensional case, and in [95] in three dimensions.”
- Unified finite element discretizations of coupled Darcy–Stokes flow (2009, unknown): “However, a three dimensional analog of the non–conforming finite element space of [17] is discussed in [23].”