Global regularity properties of steady shear thinning flows

In this paper we study the regularity of weak solutions to systems of p-Stokes type, describing the motion of some shear thinning fluids in certain steady regimes. In particular we address the problem of regularity up to the boundary improving previous results especially in terms of the allowed range for the parameter p

The well-posedness issue for an inviscid zero-Mach number system in general Besov spaces

The present paper is devoted to the study of a zero-Mach number system with heat conduction but no viscosity. We work in the framework of general non-homogeneous Besov spaces $B^s_{p,r}(\mathbb{R}^d)$, with $p\in[2,4]$ and for any $d\geq 2$, which can be embedded into the class of globally Lipschitz functions. We prove a local in time well-posedness result in these classes for general initial densities and velocity fields. Moreover, we are able to show a continuation criterion and a lower bound for the lifespan of the solutions. The proof of the results relies on Littlewood-Paley decomposition and paradifferential calculus, and on refined commutator estimates in Chemin-Lerner spaces.Comment: This submission supersedes the first part of arXiv:1305.113

A Virtual Element Method for elastic and inelastic problems on polytope meshes

We present a Virtual Element Method (VEM) for possibly nonlinear elastic and inelastic problems, mainly focusing on a small deformation regime. The numerical scheme is based on a low-order approximation of the displacement field, as well as a suitable treatment of the displacement gradient. The proposed method allows for general polygonal and polyhedral meshes, it is efficient in terms of number of applications of the constitutive law, and it can make use of any standard black-box constitutive law algorithm. Some theoretical results have been developed for the elastic case. Several numerical results within the 2D setting are presented, and a brief discussion on the extension to large deformation problems is included

The Virtual Element Method with curved edges

In this paper we initiate the investigation of Virtual Elements with curved faces. We consider the case of a fixed curved boundary in two dimensions, as it happens in the approximation of problems posed on a curved domain or with a curved interface. While an approximation of the domain with polygons leads, for degree of accuracy $k \geq 2$, to a sub-optimal rate of convergence, we show (both theoretically and numerically) that the proposed curved VEM lead to an optimal rate of convergence

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