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Abstract We introduce a new regularization model for incompressible fluid flow, which is a regularization of the EMAC (energy, momentum, and angular momentum conserving) formulation of the Navier–Stokes equations (NSE) that we call EMAC-Reg. The EMAC formulation has proved to be a useful formulation because it conserves energy, momentum, and angular momentum even when the divergence constraint is only weakly enforced. However, it is still a NSE formulation and so cannot resolve higher Reynolds number flows without very fine meshes. By carefully introducing regularization into the EMAC formulation, we create a model more suitable for coarser mesh computations but that still conserves the same quantities as EMAC, i.e., energy, momentum, and angular momentum. We show that EMAC-Reg, when semi-discretized with a finite element spatial discretization is well-posed and optimally accurate. Numerical results are provided that show EMAC-Reg is a robust coarse mesh model.
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This content will become publicly available on May 28, 2026
Robust error analysis of stabilized linear EMAC-ESAV finite element schemes for the incompressible Navier-Stokes equations
In this paper, we propose and study first- and second-order (in time) stabilized linear finite element schemes for the incompressible Navier-Stokes (NS) equations. The energy, momentum, and angular momentum conserving (EMAC) formulation has emerged as a promising approach for conserving energy, momentum, and angular momentum of the NS equations, while the exponential scalar auxiliary variable (ESAV) has become a popular technique for designing linear energy-stable numerical schemes. Our method leverages the EMAC formulation and the Taylor-Hood element with grad-div stabilization for spatial discretization. We adopt the implicit-explicit backward differential formulas (BDFs) coupled with a novel stabilized ESAV approach for time stepping. For the solution process, we develop an efficient decoupling technique for the resulting fully-discrete systems so that only one linear Stokes solve is needed at each time step, which is similar to the cost of classic implicit-explicit BDF schemes for the NS equations. Robust optimal error estimates are successfully derived for both velocity and pressure for the two proposed schemes, with Gronwall constants that are particularly independent of the viscosity. Furthermore, it is rigorously shown that the grad-div stabilization term can greatly alleviate the viscosity-dependence of the mesh size constraint, which is required for error estimation when such a term is not present in the schemes. Various numerical experiments are conducted to verify the theoretical results and demonstrate the effectiveness and efficiency of the grad-div and ESAV stabilization strategies and their combination in the proposed numerical schemes, especially for problems with high Reynolds numbers.
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- Award ID(s):
- 2409634
- PAR ID:
- 10617925
- Publisher / Repository:
- AMS
- Date Published:
- Journal Name:
- Mathematics of Computation
- ISSN:
- 0025-5718
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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