  We present a novel fully implicit hybrid finite volume/finite element method
for incompressible flows. Following previous works on semi-implicit hybrid
FV/FE schemes, the incompressible Navier-Stokes equations are split into a
pressure and a transport-diffusion subsystem. The first of them can be seen as
a Poisson type problem and is thus solved efficiently using classical
continuous Lagrange finite elements. On the other hand, finite volume methods
are employed to solve the convective subsystem, in combination with
Crouzeix-Raviart finite elements for the discretization of the viscous stress
tensor. For some applications, the related CFL condition, even if depending
only in the bulk velocity, may yield a severe time restriction in case explicit
schemes are used. To overcome this issue an implicit approach is proposed. The
system obtained from the implicit discretization of the transport-diffusion
operator is solved using an inexact Newton-Krylov method, based either on the
BiCStab or the GMRES algorithm. To improve the convergence properties of the
linear solver a symmetric Gauss-Seidel (SGS) preconditioner is employed,
together with a simple but efficient approach for the reordering of the grid
elements that is compatible with MPI parallelization. Besides, considering the
Ducros flux for the nonlinear convective terms we can prove that the discrete
advection scheme is kinetic energy stable. The methodology is carefully
assessed through a set of classical benchmarks for fluid mechanics. A last test
shows the potential applicability of the method in the context of blood flow
simulation in realistic vessel geometries.
