High Order Hierarchical Divergence-free Constrained Transport $H(div)$ Finite Element Method for Magnetic Induction Equation
| Autor: | Wei Cai, Jun Hu, Shangyou Zhang |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: |
Control and Optimization
Basis (linear algebra) Applied Mathematics Mathematical analysis Scalar (mathematics) 010103 numerical & computational mathematics Numerical Analysis (math.NA) 16. Peace & justice 01 natural sciences Finite element method Magnetic field Electromagnetic induction 010101 applied mathematics Computational Mathematics Modeling and Simulation FOS: Mathematics Constant function Mathematics - Numerical Analysis 0101 mathematics 65M60 76W05 Constant (mathematics) Divergence (statistics) Mathematics |
| Popis: | 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. arXiv admin note: text overlap with arXiv:1210.5575 |
| Databáze: | OpenAIRE |
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