A fully-discrete virtual element method for the nonstationary Boussinesq equations

TL;DR

This paper introduces a high-order fully-discrete virtual element method for solving the nonstationary Boussinesq equations, combining conforming virtual element approaches with backward Euler time discretization, and provides theoretical analysis and numerical validation.

Contribution

It presents a novel high-order virtual element method for the nonstationary Boussinesq system with proven stability, error estimates, and benchmark validation.

Findings

Unconditional stability of the scheme is established.

Error estimates in $H^2$ and $H^1$ norms are derived.

Numerical tests confirm theoretical error bounds.

Abstract

In the present work we propose and analyze a fully coupled virtual element method of high order for solving the two dimensional nonstationary Boussinesq system in terms of the stream-function and temperature fields. The discretization for the spatial variables is based on the coupling $C^1$- and $C^0$-conforming virtual element approaches, while a backward Euler scheme is employed for the temporal variable. Well-posedness and unconditional stability of the fully-discrete problem is provided. Moreover, error estimates in $H^2$- and $H^1$-norms are derived for the stream-function and temperature, respectively. Finally, a set of benchmark tests are reported to confirm the theoretical error bounds and illustrate the behavior of the fully-discrete scheme.