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.

翻译：我们提出了一种用于不可压缩流体的新型全隐式混合有限体积/有限元方法。借鉴半隐式混合FV/FE方案的先前工作，不可压缩Navier-Stokes方程被分解为压力子系统和输运-扩散子系统。前者可视为泊松型问题，因此采用经典连续拉格朗日有限元高效求解；而输运子系统则采用有限体积方法求解，并结合Crouzeix-Raviart有限元对粘性应力张量进行离散。在某些应用中，即使CFL条件仅依赖于体速度，若采用显式格式仍可能造成严格的时间步长限制。为克服此问题，本文提出一种隐式方法。通过对输运-扩散算子进行隐式离散获得的系统，采用基于BiCStab或GMRES算法的不精确Newton-Krylov方法求解。为改善线性求解器的收敛性，我们采用对称高斯-赛德尔预处理器，并结合一种简单高效且兼容MPI并行化的网格单元重排序策略。此外，通过采用Ducros通量处理非线性对流项，可证明该离散对流格式具有动能稳定性。该方法通过一系列经典流体力学基准算例进行了严格验证。最后一项测试展示了该方法在真实血管几何结构血流模拟中的潜在应用价值。