# High Order Hierarchical Divergence-free Constrained Transport $H(div)$ Finite Element Method for Magnetic Induction Equation

## TL;DR

This paper introduces a high-order hierarchical divergence-free finite element method for the magnetic induction equation, improving the enforcement of the divergence-free condition of magnetic fields in MHD simulations.

## Contribution

It proposes a novel hierarchical basis approach using interior functions of BDM_p elements to enforce divergence-free magnetic fields in finite element solutions.

## Key findings

- Effective enforcement of divergence-free condition in 3D magnetic induction simulations
- High-order accuracy achieved with hierarchical basis functions
- Numerical results demonstrate improved solution quality

## Abstract

In this paper, we will use the interior functions of an hierarchical basis for high order $BDM_p$ elements to enforce the divergence-free condition of a magnetic field $B$ approximated by the H(div) $BDM_p$ basis. The resulting constrained finite element method can be used to solve magnetic induction equation in MHD equations. The proposed procedure is based on the fact that the scalar $(p-1)$-th order polynomial space on each element can be decomposed as an orthogonal sum of the subspace defined by the divergence of the interior functions of the $p$-th order $BDM_p$ basis and the constant function. Therefore, the interior functions can be used to remove element-wise all higher order terms except the constant in the divergence error of the finite element solution of $B$-field. The constant terms from each element can be then easily corrected using a first order H(div) basis globally. Numerical results for a 3-D magnetic induction equation show the effectiveness of the proposed method in enforcing divergence-free condition of the magnetic field.