A stabilized virtual element framework for the steady state Boussinesq equation with temperature-dependent parameters

TL;DR

This paper introduces a new stabilized virtual element method for the steady state Boussinesq equation with temperature-dependent parameters, improving implementation simplicity and robustness.

Contribution

A conforming stabilized virtual element framework for the Boussinesq equation with temperature-dependent properties, avoiding higher-order derivatives and reducing degrees of freedom.

Findings

Method is more straightforward to implement.

Provides stable and accurate solutions.

Numerical tests confirm theoretical error estimates.

Abstract

This work presents a new conforming stabilized virtual element method for the generalized Boussinesq equation with temperature-dependent viscosity and thermal conductivity. A gradient-based local projection stabilization method is introduced in the discrete formulation to circumvent the violation of the discrete inf-sup condition. The well-posedness of the continuous problem is established under sufficiently small datum. We derive a stabilized virtual element problem for the Boussinesq equation using equal-order virtual element approximations. The proposed method has several advantages, such as being more straightforward to implement, free from higher-order derivative terms, providing separate stabilization terms without introducing coupling between solution components, and minimizing the number of globally coupled degrees of freedom. The existence of a discrete solution to the stabilized virtual element problem is demonstrated using the Brouwer fixed-point theorem. The error estimates are derived in the energy norm. Additionally, several numerical examples are presented to show the efficiency and robustness of the proposed method, confirming the theoretical results.