Conservative nonconforming virtual element method for stationary incompressible magnetohydrodynamics

TL;DR

This paper introduces a conservative nonconforming virtual element method for stationary incompressible magnetohydrodynamics, ensuring mass conservation, stability, and optimal error estimates, validated through numerical experiments.

Contribution

It develops a novel virtual element method that guarantees divergence-free velocity fields and provides optimal error bounds without extra conditions.

Findings

Ensures mass conservation for velocity field.

Achieves optimal error estimates in energy and L^2 norms.

Validates theoretical results with numerical experiments.

Abstract

In this paper, we propose a conservative nonconforming virtual element method for the full stationary incompressible magnetohydrodynamics model. We leverage the virtual element satisfactory divergence-free property to ensure mass conservation for the velocity field. The condition of the well-posedness of the proposed method, as well as the stability are derived. We establish optimal error estimates in the discrete energy norm for both the velocity and magnetic field. Furthermore, by employing a new technique, we obtain the optimal error estimates in $L^2$-norm without any additional conditions. Finally, numerical experiments are presented to validate the theoretical analysis. In the implementation process, we adopt the effective Oseen iteration to handle the nonlinear system.