On the convergence of a low order Lagrange finite element approach for natural convection problems

TL;DR

This paper proves the convergence of a low order finite element method for natural convection problems, demonstrating its effectiveness and computational efficiency with theoretical and numerical validation.

Contribution

It establishes convergence conditions for a P1 finite element approach with a penalty term, enabling reliable simulation of natural convection systems.

Findings

Convergence is proven under specific penalty and regularity assumptions.

Optimal order of convergence can be achieved with proper parameter choices.

The method reduces computational costs compared to higher-order approaches.

Abstract

The purpose of this article is to study the convergence of a low order finite element approximation for a natural convection problem. We prove that the discretization based on P1 polynomials for every variable (velocity, pressure and temperature) is well-posed if used with a penalty term in the divergence equation, to compensate the loss of an inf-sup condition. With mild assumptions on the pressure regularity, we recover convergence for the Navier-Stokes-Boussinesq system, provided the penalty term is chosen in accordance with the mesh size. We express conditions to obtain optimal order of convergence. We illustrate theoretical convergence results with extensive examples. The computational cost that can be saved by this approach is also assessed.