766 research outputs found
Virtual Elements for the Navier-Stokes problem on polygonal meshes
A family of Virtual Element Methods for the 2D Navier-Stokes equations is
proposed and analysed. The schemes provide a discrete velocity field which is
point-wise divergence-free. A rigorous error analysis is developed, showing
that the methods are stable and optimally convergent. Several numerical tests
are presented, confirming the theoretical predictions. A comparison with some
mixed finite elements is also performed
An advection-robust Hybrid High-Order method for the Oseen problem
In this work, we study advection-robust Hybrid High-Order discretizations of
the Oseen equations. For a given integer , the discrete velocity
unknowns are vector-valued polynomials of total degree on mesh elements
and faces, while the pressure unknowns are discontinuous polynomials of total
degree on the mesh. From the discrete unknowns, three relevant
quantities are reconstructed inside each element: a velocity of total degree
, a discrete advective derivative, and a discrete divergence. These
reconstructions are used to formulate the discretizations of the viscous,
advective, and velocity-pressure coupling terms, respectively. Well-posedness
is ensured through appropriate high-order stabilization terms. We prove energy
error estimates that are advection-robust for the velocity, and show that each
mesh element of diameter contributes to the discretization error with
an -term in the diffusion-dominated regime, an
-term in the advection-dominated regime, and
scales with intermediate powers of in between. Numerical results complete
the exposition
p- and hp- virtual elements for the Stokes problem
We analyse the p- and hp-versions of the virtual element method (VEM) for the
the Stokes problem on a polygonal domain. The key tool in the analysis is the
existence of a bijection between Poisson-like and Stokes-like VE spaces for the
velocities. This allows us to re-interpret the standard VEM for Stokes as a
VEM, where the test and trial discrete velocities are sought in Poisson-like VE
spaces. The upside of this fact is that we inherit from [7] an explicit
analysis of best interpolation results in VE spaces, as well as stabilization
estimates that are explicit in terms of the degree of accuracy of the method.
We prove exponential convergence of the hp-VEM for Stokes problems with regular
right-hand sides. We corroborate the theoretical estimates with numerical tests
for both the p- and hp-versions of the method
Divergence-preserving reconstructions on polygons and a really pressure-robust virtual element method for the Stokes problem
Non divergence-free discretisations for the incompressible Stokes problem may suffer from a lack of pressure-robustness characterised by large discretisations errors due to irrotational forces in the momentum balance. This paper argues that also divergence-free virtual element methods (VEM) on polygonal meshes are not really pressure-robust as long as the right-hand side is not discretised in a careful manner. To be able to evaluate the right-hand side for the testfunctions, some explicit interpolation of the virtual testfunctions is needed that can be evaluated pointwise everywhere. The standard discretisation via an L2 -bestapproximation does not preserve the divergence and so destroys the orthogonality between divergence-free testfunctions and possibly eminent gradient forces in the right-hand side. To repair this orthogonality and restore pressure-robustness another divergence-preserving reconstruction is suggested based on Raviart--Thomas approximations on local subtriangulations of the polygons. All findings are proven theoretically and are demonstrated numerically in two dimensions. The construction is also interesting for hybrid high-order methods on polygonal or polyhedral meshes
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