We present a novel asymptotic-preserving semi-implicit finite element method for weakly compressible and incompressible flows based on compatible finite element spaces. The momentum is sought in an $H(\mathrm{div})$-conforming space, ensuring exact pointwise mass conservation at the discrete level. We use an explicit discontinuous Galerkin-based discretization for the convective terms, while treating the pressure and viscous terms implicitly, so that the CFL condition depends only on the fluid velocity. To handle shocks and damp spurious oscillations in the compressible regime, we incorporate an a posteriori limiter that employs artificial viscosity and is based on a discrete maximum principle. By using hybridization, the final algorithm requires solving only symmetric positive definite linear systems. As the Mach number approaches zero and the density remains constant, the method converges to an $H(\mathrm{div})$-based discretization of the incompressible Navier-Stokes equations in the vorticity-velocity-pressure formulation. Several numerical tests validate the proposed method.

翻译：本文提出了一种基于相容有限元空间、适用于弱可压缩与不可压缩流动的新型渐近保持半隐式有限元方法。动量方程在$H(\mathrm{div})$-相容空间中求解，确保在离散层面实现精确的逐点质量守恒。对流项采用显式间断伽辽金离散，而压力项与粘性项则进行隐式处理，使得CFL条件仅取决于流体速度。为处理可压缩区域中的激波并抑制伪振荡，我们引入了一种基于离散极值原理、采用人工粘性的后验限制器。通过杂交化技术，最终算法仅需求解对称正定线性系统。当马赫数趋近于零且密度保持恒定，该方法收敛至基于$H(\mathrm{div})$的涡量-速度-压力形式不可压缩Navier-Stokes方程离散格式。多个数值算例验证了所提方法的有效性。