TY - JOUR
T1 - An Implicit Staggered Hybrid Finite Volume/Finite Element Solver for the Incompressible Navier-Stokes Equations
AU - Lucca , Alessia
AU - Busto , Saray
AU - Dumbser , Michael
JO - East Asian Journal on Applied Mathematics
VL - 3
SP - 671
EP - 716
PY - 2023
DA - 2023/05
SN - 13
DO - http://doi.org/10.4208/eajam.2022-335.300123 
UR - https://global-sci.org/intro/article_detail/eajam/21726.html
KW - Hybrid finite volume/finite element method, finite volume scheme, continuous finite element method, incompressible Navier-Stokes equations for blood flow applications, staggered implicit schemes.
AB - <p style="text-align: justify;">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.</p>