Parametric charge-conservative mixed finite element method for 3D incompressible inductionless MHD equations on curved domains

TL;DR

This paper introduces a charge-conservative mixed finite element method for 3D incompressible inductionless MHD equations on curved domains, achieving optimal convergence and divergence-free current density.

Contribution

It develops a novel finite element scheme that ensures charge conservation and divergence-free current density on complex 3D curved geometries.

Findings

Achieves optimal convergence rates in energy and L2 norms.

Ensures exactly divergence-free current density via Piola's transformation.

Numerical results confirm theoretical error estimates.

Abstract

This paper develops a charge-conservative mixed finite element method with optimal convergence rates for the stationary incompressible inductionless MHD equations on three-dimensional curved domains. The discretization employs the isoparametric Taylor-Hood elements with grad-div stabilization for the velocity-pressure pair, and parametric Brezzi-Douglas-Marini elements for the current density. Utilizing the Piola's transformation, the discrete current density is exactly divergence-free. By employing suitable extensions and projections, optimal a priori error estimates are derived in both the energy norm and the $L^2$-norm. Numerical experiments are presented to confirm the theoretical results.