2,338 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
A parallel multigrid solver for multi-patch Isogeometric Analysis
Isogeometric Analysis (IgA) is a framework for setting up spline-based
discretizations of partial differential equations, which has been introduced
around a decade ago and has gained much attention since then. If large spline
degrees are considered, one obtains the approximation power of a high-order
method, but the number of degrees of freedom behaves like for a low-order
method. One important ingredient to use a discretization with large spline
degree, is a robust and preferably parallelizable solver. While numerical
evidence shows that multigrid solvers with standard smoothers (like Gauss
Seidel) does not perform well if the spline degree is increased, the multigrid
solvers proposed by the authors and their co-workers proved to behave optimal
both in the grid size and the spline degree. In the present paper, the authors
want to show that those solvers are parallelizable and that they scale well in
a parallel environment.Comment: The first author would like to thank the Austrian Science Fund (FWF)
for the financial support through the DK W1214-04, while the second author
was supported by the FWF grant NFN S117-0
SUPG-stabilized Virtual Elements for diffusion-convection problems: a robustness analysis
The objective of this contribution is to develop a convergence analysis for
SUPG-stabilized Virtual Element Methods in diffusion-convection problems that
is robust also in the convection dominated regime. For the original method
introduced in [Benedetto et al, CMAME 2016] we are able to show an "almost
uniform" error bound (in the sense that the unique term that depends in an
unfavorable way on the parameters is damped by a higher order mesh-size
multiplicative factor). We also introduce a novel discretization of the
convection term that allows us to develop error estimates that are fully robust
in the convection dominated cases. We finally present some numerical result
Equilibrium analysis of an immersed rigid leaflet by the virtual element method
We study, both theoretically and numerically, the equilibrium of a hinged rigid leaflet with an attached rotational spring, immersed in a stationary incompressible fluid within a rigid channel. Through a careful investigation of the properties of the domain functional describing the angular momentum exerted by the fluid on the leaflet (which depends on both the leaflet angular position and its thickness), we identify sufficient conditions on the spring stiffness function for the existence (and uniqueness) of equilibrium positions. This study resorts to techniques from shape differential calculus. We propose a numerical technique that exploits the mesh flexibility of the Virtual Element Method (VEM). A (polygonal) computational mesh is generated by cutting a fixed background grid with the leaflet geometry, and the problem is then solved with stable VEM Stokes elements of degrees 1 and 2 combined with a bisection algorithm. We prove quasi-optimal error estimates and present a large array of numerical experiments to document the accuracy and robustness with respect to degenerate geometry of the proposed methodology
Weak in Space, Log in Time Improvement of the Lady{\v{z}}enskaja-Prodi-Serrin Criteria
In this article we present a Lady{\v{z}}enskaja-Prodi-Serrin Criteria for
regularity of solutions for the Navier-Stokes equation in three dimensions
which incorporates weak norms in the space variables and log improvement
in the time variable.Comment: 14 pages, to appea
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