We propose a pressure-robust enriched Galerkin (EG) finite element method for the incompressible Navier-Stokes and heat equations in the Boussinesq regime. For the Navier-Stokes equations, the EG formulation combines continuous Lagrange elements with a discontinuous enrichment vector per element in the velocity space and a piecewise constant pressure space, and it can be implemented efficiently within standard finite element frameworks. To enforce pressure robustness, we construct velocity reconstruction operators that map the discrete EG velocity field into exactly divergence-free, H(div)-conforming fields. In particular, we develop reconstructions based on Arbogast-Correa (AC) mixed finite element spaces on quadrilateral meshes and demonstrate that the resulting schemes remain stable and accurate even on highly distorted grids. The nonlinearity of the coupled Navier-Stokes-Boussinesq system is treated with several iterative strategies, including Picard iterations and Anderson-accelerated iterations; our numerical study shows that Anderson acceleration yields robust and efficient convergence for high Rayleigh number flows within the proposed framework. The performance of the method is assessed on a set of benchmark problems and application-driven test cases. These numerical experiments highlight the potential of pressure-robust EG methods as flexible and accurate tools for coupled flow and heat transport in complex geometries.

翻译：本文针对Boussinesq近似下的不可压缩Navier-Stokes方程与热方程，提出一种压力鲁棒的增广Galerkin（EG）有限元方法。对于Navier-Stokes方程，EG格式在速度空间中采用连续Lagrange单元与每个单元上的间断增广向量相结合，压力空间采用分片常数函数，该格式可在标准有限元框架内高效实现。为实现压力鲁棒性，我们构造了速度重构算子，将离散EG速度场映射为严格散度为零且满足H(div)相容性的场。特别地，基于四边形网格上的Arbogast-Correa（AC）混合有限元空间开发了重构算子，并证明所得格式即使在高度扭曲网格上仍保持稳定与精确。针对耦合Navier-Stokes-Boussinesq系统的非线性问题，采用多种迭代策略处理，包括Picard迭代与Anderson加速迭代；数值研究表明，在本文提出的框架下，Anderson加速对高Rayleigh数流动能实现鲁棒且高效的收敛。通过一系列基准问题与应用驱动测试案例评估了方法的性能。数值实验凸显了压力鲁棒EG方法作为复杂几何中流动与热输运耦合问题灵活且精确求解工具的潜力。