A stabilized finite element method based on two local Gauss integrations for the Stokes equations
Journal of Computational and Applied Mathematics
Open and traction boundary conditions for the incompressible Navier-Stokes equations
Journal of Computational Physics
Applied Numerical Mathematics
Stable and accurate pressure approximation for unsteady incompressible viscous flow
Journal of Computational Physics
Journal of Computational Physics
Applied Numerical Mathematics
Journal of Computational and Applied Mathematics
Consistent Local Projection Stabilized Finite Element Methods
SIAM Journal on Numerical Analysis
SIAM Journal on Scientific Computing
Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics
Computers & Mathematics with Applications
Stabilization and scalable block preconditioning for the Navier-Stokes equations
Journal of Computational Physics
HPCA'09 Proceedings of the Second international conference on High Performance Computing and Applications
Two-level stabilized method based on three corrections for the stationary Navier-Stokes equations
Applied Numerical Mathematics
Mathematics and Computers in Simulation
SIAM Journal on Numerical Analysis
On the semi-discrete stabilized finite volume method for the transient Navier---Stokes equations
Advances in Computational Mathematics
Mixed mimetic spectral element method for Stokes flow: A pointwise divergence-free solution
Journal of Computational Physics
Journal of Computational and Applied Mathematics
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We present a new family of stabilized methods for the Stokes problem. The focus of the paper is on the lowest order velocity-pressure pairs. While not LBB compliant, their simplicity and attractive computational properties make these pairs a popular choice in engineering practice. Our stabilization approach is motivated by terms that characterize the LBB "deficiency" of the unstable spaces. The stabilized methods are defined by using these terms to modify the saddle-point Lagrangian associated with the Stokes equations. The new stabilized methods offer a number of attractive computational properties. In contrast to other stabilization procedures, they are parameter free, do not require calculation of higher order derivatives or edge-based data structures, and always lead to symmetric linear systems. Furthermore, the new methods are unconditionally stable, achieve optimal accuracy with respect to solution regularity, and have simple and straightforward implementations. We present numerical results in two and three dimensions that showcase the excellent stability and accuracy of the new methods.