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

Sanjeeb Poudel, Sanghyun Lee, Lin Mu

TL;DR

The paper develops a pressure-robust enriched Galerkin framework for the coupled incompressible Navier–Stokes and heat equations under the Boussinesq approximation. It achieves pressure robustness through a velocity reconstruction to divergence-free fields using Arbogast–Correa elements on quadrilaterals, enabling stability on distorted meshes. Anderson-accelerated Picard iterations are employed to robustly solve the nonlinear system, particularly at high Rayleigh numbers. Numerical experiments on natural convection benchmarks, distorted meshes, and porous-media heat transfer demonstrate accurate, Reynolds-number–independent velocity errors and strong fidelity to reference solutions, highlighting practical potential for complex geometries.

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.

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

TL;DR

The paper develops a pressure-robust enriched Galerkin framework for the coupled incompressible Navier–Stokes and heat equations under the Boussinesq approximation. It achieves pressure robustness through a velocity reconstruction to divergence-free fields using Arbogast–Correa elements on quadrilaterals, enabling stability on distorted meshes. Anderson-accelerated Picard iterations are employed to robustly solve the nonlinear system, particularly at high Rayleigh numbers. Numerical experiments on natural convection benchmarks, distorted meshes, and porous-media heat transfer demonstrate accurate, Reynolds-number–independent velocity errors and strong fidelity to reference solutions, highlighting practical potential for complex geometries.

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.

Paper Structure

This paper contains 18 sections, 1 theorem, 63 equations, 19 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Governing System
  3. Numerical Algorithm
  4. Temporal Discretization
  5. Picard Iteration
  6. Anderson-accelerated Picard iteration (AA--Picard)
  7. Spatial Discretization
  8. Enriched Galerkin Finite Element Method
  9. Arbogast-Correa (AC) elements
  10. Pressure-Robust Enhancement with AC element
  11. Mass Conservation
  12. Numerical Experiments
  13. Example 1. Natural convection in a square cavity
  14. Example 2. Convergence of a smooth solution with high Reynolds number
  15. Homogeneous boundary case
  16. ...and 3 more sections

Key Result

Theorem 1

For the numerical solution $({\bf u}_n,p_h)$ obtained from Eqn. eq:pr-ac-scheme, we have the following mass conservation

Figures (19)

  • Figure 1: Reference rectangle element $\hat{T}$ and physical trapezoid element $T$.
  • Figure 2: An example of the AC$_0$ basis functions: vector plot in a trapezoid element with four vertices $\{(0,0),(2,0),(1,2),(0,2)\}$.
  • Figure 3: Pressure distribution and velocity field at $t_f = 1$ for various Ra numbers ($\text{Ra}$).
  • Figure 4: Temperature distribution at $t_f = 1$ for various Rayleigh number.
  • Figure 5: Comparison of the convergence of Picard and AA-Picard (with $m = 10$) for different Rayleigh numbers.
  • ...and 14 more figures

Theorems & Definitions (3)

  • Remark 1
  • Remark 2
  • Theorem 1