A full divergence-free of high order virtual finite element method to approximation of stationary inductionless magnetohydrodynamic equations on polygonal meshes

Abstract

In this present paper we consider a full divergence-free of high order virtual finite element algorithm to approximate the stationary inductionless magnetohydrodynamic model on polygonal meshes. More precisely, we choice appropriate virtual spaces and necessary degrees of freedom for velocity and current density to guarantee that their final discrete formats are both pointwise divergence-free. Moreover, we hope to achieve higher approximation accuracy at higher "polynomial" orders k_{1} \geq 2, k_{2} \geq 1, while the full divergence-free property has always been satisfied. And then we processed rigorous error analysis to show that the proposed method is stable and convergent. Several numerical tests are presented, confirming the theoretical predictions.