A new family of mixed finite elements in IR3
Numerische Mathematik
Piecewise solenoidal vector fields and the Stokes problem
SIAM Journal on Numerical Analysis
A Nonconforming Finite Element Method for the Stationary Navier--Stokes Equations
SIAM Journal on Numerical Analysis
A high-order discontinuous Galerkin method for 2D incompressible flows
Journal of Computational Physics
Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
Journal of Scientific Computing
Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems
SIAM Journal on Numerical Analysis
Local Discontinuous Galerkin Methods for the Stokes System
SIAM Journal on Numerical Analysis
Incompressible Finite Elements via Hybridization. Part I: The Stokes System in Two Space Dimensions
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
A high-order discontinuous Galerkin method for the unsteady incompressible Navier-Stokes equations
Journal of Computational Physics
Journal of Computational Physics
A Note on Discontinuous Galerkin Divergence-free Solutions of the Navier-Stokes Equations
Journal of Scientific Computing
New Finite Element Methods in Computational Fluid Dynamics by H(div) Elements
SIAM Journal on Numerical Analysis
Journal of Computational Physics
Journal of Computational Physics
The Derivation of Hybridizable Discontinuous Galerkin Methods for Stokes Flow
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
SIAM Journal on Scientific Computing
An Analysis of the Embedded Discontinuous Galerkin Method for Second-Order Elliptic Problems
SIAM Journal on Numerical Analysis
A Hybridizable Discontinuous Galerkin Method for Steady-State Convection-Diffusion-Reaction Problems
SIAM Journal on Scientific Computing
A Comparison of HDG Methods for Stokes Flow
Journal of Scientific Computing
High-order implicit hybridizable discontinuous Galerkin methods for acoustics and elastodynamics
Journal of Computational Physics
Hybridizable discontinuous Galerkin methods for the time-harmonic Maxwell's equations
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
A space-time discontinuous Galerkin method for the incompressible Navier-Stokes equations
Journal of Computational Physics
Journal of Computational Physics
Hi-index | 31.48 |
We present an implicit high-order hybridizable discontinuous Galerkin method for the steady-state and time-dependent incompressible Navier-Stokes equations. The method is devised by using the discontinuous Galerkin discretization for a velocity gradient-pressure-velocity formulation of the incompressible Navier-Stokes equations with a special choice of the numerical traces. The method possesses several unique features which distinguish itself from other discontinuous Galerkin methods. First, it reduces the globally coupled unknowns to the approximate trace of the velocity and the mean of the pressure on element boundaries, thereby leading to a significant reduction in the degrees of freedom. Moreover, if the augmented Lagrangian method is used to solve the linearized system, the globally coupled unknowns become the approximate trace of the velocity only. Second, it provides, for smooth viscous-dominated problems, approximations of the velocity, pressure, and velocity gradient which converge with the optimal order of k+1 in the L^2-norm, when polynomials of degree k=0 are used for all components of the approximate solution. And third, it displays superconvergence properties that allow us to use the above-mentioned optimal convergence properties to define an element-by-element postprocessing scheme to compute a new and better approximate velocity. Indeed, this new approximation is exactly divergence-free, H(div)-conforming, and converges with order k+2 for k=1 and with order 1 for k=0 in the L^2-norm. Moreover, a novel and systematic way is proposed for imposing boundary conditions for the stress, viscous stress, vorticity and pressure which are not naturally associated with the weak formulation of the method. This can be done on different parts of the boundary and does not result in the degradation of the optimal order of convergence properties of the method. Extensive numerical results are presented to demonstrate the convergence and accuracy properties of the method for a wide range of Reynolds numbers and for various polynomial degrees.