A Note on Discontinuous Galerkin Divergence-free Solutions of the Navier–Stokes Equations

By Bernardo Cockburn, Guido Kanschat, Dominik Schötzau. Journal of Scientific Computing, 2007

305 citations recorded by the source corpus.

Subfields: Computational Mechanics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/72b2a41c57d3fe10e88c796f6005dba4f3ea1ecb

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Citations in the corpus

A family of nonconforming elements for the Brinkman problem (2012, unknown): “We can also choose q so that 〈[y · n], ω〉F = 0 for all ω ∈ P k−1(F ) and all faces F .”
On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime (2014, methodology): “This price, which by the way can be avoided completely by divergence-free mixed methods like the classical Scott-Vogelius element[49, 48, 41, 51] or more recent divergence-free discretization approaches[11, 12, 50, 20, 21, 22], is well-known in the finite element community as poor mass conservation ”
Discontinuous Galerkin methods for the Stokes equations using divergence‐free approximations (2008, background): “More recently, Cockburn and coworkers propose [6, 9, 10 ] a DG formulation with solenoidal piecewise polynomial approximations.”
IsoGeometric Analysis: Stable elements for the 2D Stokes equation (2011, background): “For that reason classical (discontinuous) Raviart–Thomas and Nédélec elements are not popular in the finite element approximations of the Stokes problem, since they lead to non-conforming schemes, though interesting results are proposed in [9, 15, 16].”
A mixed finite element method with exactly divergence-free velocities for incompressible magnetohydrodynamics (2010, methodology): “We note that divergence-conforming discretizations have been analyzed for the incompressible Navier– Stokes equations in [11].”
H(div)-conforming Finite Elements for the Brinkman Problem (2011, background): “H(div)elements have been considered for the closely related Stokes problem in [9, 26, 14].”
On the Divergence Constraint in Mixed Finite Element Methods for Incompressible Flows (2015, methodology): “Here, the symmetric interior penalty arguments given in [65, 64] are presented, although different discontinuous Galerkin techniques are available [20].”
Robust Globally Divergence-Free Weak Galerkin Methods for Stokes Equations (2016, methodology): “It has been shown in [25, 62] that DG methods using H(div)-conforming elements lead to divergence-free approximations.”
Divergence‐free discontinuous Galerkin schemes for the Stokes equations and the MAC scheme (2008, methodology): “1 Introduction Divergence-conforming discontinuous Galerkin schemes recently suggested by Cockburn, Kanschat and Schötzau (see [2, 3], abbreviated as CKS below) generate pointwise divergence free solutions to the incompressible Navier-Stokes equations without using a solenoidal basis; therefore, the”
ISOGEOMETRIC DIVERGENCE-CONFORMING B-SPLINES FOR THE STEADY NAVIER–STOKES EQUATIONS (2013, background): “Alternatively, provably stable, convergent, and locally-conservative discontinuous Galerkin discretizations have been devised for the steady Navier-Stokes equations in [11, 12], but these discretizations are encumbered with a proliferation of degrees of freedom and are thus largely limited to two sp”