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

TL;DR

This paper introduces a pressure-robust enriched Galerkin finite element method for coupled Navier-Stokes and heat equations, ensuring stability and accuracy on distorted grids and demonstrating robustness with iterative solvers in high Rayleigh number flows.

Contribution

The paper develops a novel pressure-robust EG finite element scheme with divergence-free velocity reconstruction for coupled flow and heat transfer problems.

Findings

Method remains stable and accurate on distorted grids.

Anderson acceleration improves convergence for high Rayleigh numbers.

Numerical experiments validate the method's effectiveness.

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.