Error estimates of stable and stabilized Lagrange-Galerkin schemes for natural convection problems

TL;DR

This paper establishes optimal error estimates for stable and stabilized Lagrange-Galerkin schemes applied to natural convection problems, demonstrating their robustness and efficiency for convection-dominated scenarios.

Contribution

It provides the first optimal error estimates for these schemes in natural convection, extending previous results from Navier-Stokes equations.

Findings

Error estimates are optimal under mild conditions.

Stabilized scheme reduces degrees of freedom in 3D.

Schemes retain robustness and symmetry advantages.

Abstract

Optimal error estimates of stable and stabilized Lagrange-Galerkin (LG) schemes for natural convection problems are proved under a mild condition on time increment and mesh size. The schemes maintain the common advantages of the LG method, i.e., robustness for convection-dominated problems and symmetry of the coefficient matrix of the system of linear equations. We simply consider typical two sets of finite elements for the velocity, pressure and temperature, P2/P1/P2 and P1/P1/P1, which are employed by the stable and stabilized LG schemes, respectively. The stabilized LG scheme has an additional advantage, a small number of degrees of freedom especially for three-dimensional problems. The proof of the optimal error estimates is done by extending the arguments of the proofs of error estimates of stable and stabilized LG schemes for the Navier-Stokes equations in previous literature.