Pressure-robust enriched Galerkin finite element methods for coupled Navier-Stokes and heat equations

2025-12-18

AI Summary

For the coupled Boussinesq-type incompressible Navier–Stokes and heat equations, conventional methods often suffer from pressure-induced accuracy degradation on highly distorted meshes due to lack of pressure robustness. This work proposes a pressure-robust augmented Galerkin finite element method: it employs the Arbogast–Correa mixed element to construct an H(div)-conforming velocity reconstruction operator with exactly divergence-free property, thereby ensuring intrinsic pressure robustness; and it introduces Anderson acceleration into the Picard iteration framework for the first time, markedly enhancing convergence speed and robustness for high-Rayleigh-number flows (up to $10^6$). Numerical experiments demonstrate that the method maintains high accuracy and strong stability across both benchmark and complex-geometry test cases. The proposed approach provides a novel discretization tool for fluid–thermal coupling simulations, combining geometric flexibility with high-fidelity solution quality.

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Abstract

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.

Problem

• Develops pressure-robust enriched Galerkin methods for Navier-Stokes and heat equations.
• Ensures accuracy on distorted grids via velocity reconstruction operators.
• Addresses nonlinear coupling with efficient iterative strategies for high Rayleigh flows.

Innovation

• Enriched Galerkin method with discontinuous velocity enrichment
• Pressure robustness via divergence-free velocity reconstruction operators
• Anderson-accelerated iterations for high Rayleigh number flows