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

AbstractIn this paper, we propose to use the interior functions of an hierarchical basis for high order BDMpelements to enforce the divergence-free condition of a magnetic field Bapproximated by the H(div)BDMpbasis. 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 BDMpbasis 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 the 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.