A Superconvergent HDG Method for the Maxwell Equations |
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Authors: | Huangxin Chen Weifeng Qiu Ke Shi Manuel Solano |
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Affiliation: | 1.School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing,Xiamen University,Fujian,China;2.Department of Mathematics,City University of Hong Kong,Kowloon,Hong Kong, China;3.Department of Mathematics and Statistics,Old Dominion University,Norfolk,USA;4.Departamento de Ingeniería Matemática and Centro de Investigación en Ingeniería Matemática (CI2MA),Universidad de Concepción,Concepción,Chile |
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Abstract: | We present and analyze a new hybridizable discontinuous Galerkin (HDG) method for the steady state Maxwell equations. In order to make the problem well-posed, a condition of divergence is imposed on the electric field. Then a Lagrange multiplier p is introduced, and the problem becomes the solution of a mixed curl–curl formulation of the Maxwell’s problem. We use polynomials of degree \(k+1\), k, k to approximate \({{\varvec{u}}},\nabla \times {{\varvec{u}}}\) and p respectively. In contrast, we only use a non-trivial subspace of polynomials of degree \(k+1\) to approximate the numerical tangential trace of the electric field and polynomials of degree \(k+1\) to approximate the numerical trace of the Lagrange multiplier on the faces. On the simplicial meshes, we show that the convergence rates for \(\varvec{u}\) and \(\nabla \times \varvec{u}\) are independent of the Lagrange multiplier p. If we assume the dual operator of the Maxwell equation on the domain has adequate regularity, we show that the convergence rate for \(\varvec{u}\) is \(O(h^{k+2})\). From the point of view of degrees of freedom of the globally coupled unknown: numerical trace, this HDG method achieves superconvergence for the electric field without postprocessing. Finally, we show that the superconvergence of the HDG method is also derived on general polyhedral elements. Numerical results are given to verify the theoretical analysis. |
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