A locally conservative LDG method for the incompressible Navier-Stokes equations

By Bernardo Cockburn, Guido Kanschat, Dominik Schötzau. Mathematics of Computation, 2004

331 citations recorded by the source corpus.

In this paper a new local discontinuous Galerkin method for the incompressible stationary Navier-Stokes equations is proposed and analyzed. Four important features render this method unique: its stability, its local conservativity, its high-order accuracy, and the exact satisfaction of the incompressibility constraint. Although the method uses completely discontinuous approximations, a globally divergence-free approximate velocity in H(div; Ω) is obtained by simple, element-by-element post-processing. Optimal error estimates are proven and an iterative procedure used to compute the approximate solution is shown to converge. This procedure is nothing but a discrete version of the classical fixed point iteration used to obtain existence and uniqueness of solutions to the incompressible Navier-Stokes equations by solving a sequence of Oseen problems. Numerical results are shown which verify the theoretical rates of convergence. They also confirm the independence of the number of fixed point iterations with respect to the discretization parameters. Finally, they show that the method works well for a wide range of Reynolds numbers.

Subfields: Computational Mechanics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/c82b35e639a7fc21cb7510243dbe8af783dda227

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Citations in the corpus

A mixed finite element method with exactly divergence-free velocities for incompressible magnetohydrodynamics (2010, methodology): “1), whereby all the variables are approximated in discontinuous finite element spaces, based on existing discretizations for the Oseen and Maxwell equations [9,10,35].”
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].”
Robust Globally Divergence-Free Weak Galerkin Methods for Stokes Equations (2016, unknown): “This inconvenience was avoided by local discontinuous Galerkin (LDG) methods [15, 16, 24, 29], which have an additional property that fluxes can be eliminated locally, and hybridizable discontinuous Galerkin (HDG) methods [19–23,26–28,52], which introduce the numerical trace as an unknown and posses”
A Hybrid High-Order method for the incompressible Navier-Stokes equations based on Temam's device (2018, unknown): “Readers mainly interested in the numerical recipe and results can skip this technical appendix at ﬁrst reading.”
Discontinuous Galerkin methods for incompressible elastic materials (2006, methodology): “For this very reason, the LDG method introduced in this paper is an extension of the LDG method introduced in [8] for the steady state Navier-Stokes equations.”
An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diffusion equations (2009, unknown): “We follow the postprocessing proposed in [8], which is a straightforward extension of the post-processing procedure introduced in [3,12] and later used in [7,11].”
THE DIRECT DISCONTINUOUS GALERKIN (DDG) METHOD FOR DIFFUSION WITH INTERFACE CORRECTIONS (2010, methodology): “There have been various DG methods suggested in the literature to solve the problem, including the method originally proposed by Bassi and Rebay [4] for compressible Navier-Stokes equations, its generalization called the local discontinuous Galerkin (LDG) methods introduced in [19] by Cockburn and S”
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”
A Note on Discontinuous Galerkin Divergence-free Solutions of the Navier–Stokes Equations (2007, methodology): “To describe them, let us begin by pointing out that in [15], it was shown that DG methods for the incompressible Navier-Stokes equations cannot be both locally conservative as well as energy-stable unless the approximation to the convective velocity is exactly divergence-free.”
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 ”
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”