# Robust globally divergence-free weak Galerkin finite element methods for natural convection problems

## TL;DR

This paper introduces a new class of weak Galerkin finite element methods for stationary natural convection problems, ensuring divergence-free velocity solutions and demonstrating robustness and optimal error estimates through theoretical analysis and numerical experiments.

## Contribution

The paper develops divergence-free weak Galerkin methods with proven stability, optimal error estimates, and an unconditionally convergent algorithm for natural convection problems.

## Key findings

- Methods produce divergence-free velocity fields.
- Numerical results confirm robustness across Rayleigh numbers.
- Optimal a priori error estimates are established.

## Abstract

This paper proposes and analyzes a class of weak Galerkin (WG) finite element methods for stationary natural convection problems in two and three dimensions. We use piecewise polynomials of degrees k, k-1, and k(k>=1) for the velocity, pressure, and temperature approximations in the interior of elements, respectively, and piecewise polynomials of degrees l, k, l(l = k-1,k) for the numerical traces of velocity, pressure and temperature on the interfaces of elements. The methods yield globally divergence-free velocity solutions. Well-posedness of the discrete scheme is established, optimal a priori error estimates are derived, and an unconditionally convergent iteration algorithm is presented. Numerical experiments confirm the theoretical results and show the robustness of the methods with respect to Rayleigh number.