An implicit staggered hybrid finite volume/finite element solver for the incompressible Navier-Stokes equations

TL;DR

This paper introduces a fully implicit hybrid finite volume/finite element solver for incompressible Navier-Stokes equations, improving stability and efficiency for complex fluid flow simulations including blood flow in realistic geometries.

Contribution

It develops an implicit hybrid FV/FE method with advanced linear solvers and preconditioning, enhancing stability and computational performance over semi-implicit schemes.

Findings

Proves kinetic energy stability of the discrete advection scheme.

Demonstrates the method's effectiveness on classical fluid mechanics benchmarks.

Shows potential for realistic blood flow simulations in complex geometries.

Abstract

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.