An equal-order virtual element framework for the coupled Stokes-Temperature equation with nonlinear viscosity

TL;DR

This paper introduces a new stabilized virtual element method for the coupled Stokes-Temperature equations with nonlinear viscosity, using equal-order elements on polygonal meshes, ensuring stability, optimal convergence, and robustness.

Contribution

The paper develops a novel stabilized virtual element framework for coupled Stokes-Temperature equations with nonlinear viscosity, avoiding complex derivatives and coupling issues.

Findings

Proves stability and well-posedness of the method

Achieves optimal error convergence rates

Demonstrates robustness with respect to thermal conductivity

Abstract

In this work, we present and analyze a novel stabilized virtual element formulation for the coupled Stokes-Temperature equation on polygonal meshes, employing equal-order element pairs where viscosity depends on temperature. The main objective of the proposed virtual elements is to develop a stabilized virtual element problem that avoids higher-order derivative terms and bilinear forms involving velocity, pressure and temperature, thereby avoiding the coupling between virtual element pairs. Moreover, it also reduces the violation of divergence-free constraints and offers reasonable control over the gradient of temperature. We derive the stability of the continuous solution using the Banach fixed-point theorem under sufficiently small data. The stabilized coupled virtual element problem is formulated using the local projection-based stabilization methods. We demonstrate the existence and uniqueness of the stabilized discrete solution using the Brouwer fixed-point theorem and the contraction theorem under the assumption of sufficient small data by showing the well-posedness of the stabilized decoupled virtual element problems. Furthermore, we derive the error estimates with optimal convergence rates in the energy norms. We present several numerical examples to confirm the theoretical findings. Additionally, the numerical behavior of the proposed stabilized method is shown to be robust with respect to linear and non-linear thermal conductivity.