A Mixed H1-Conforming Finite Element Method for Solving Maxwell's Equations with Non-H1 Solution
A Mixed H1-Conforming Finite Element Method for Solving Maxwell's Equations with Non-H1 Solution
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DOI：
10.1137/16m1078082
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发表时间：
2018-01
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影响因子：
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通讯作者：
Huoyuan Duan;R. C. Tan;Suh-Yuh Yang;Cheng-Shu You
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文献类型：
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作者：
Huoyuan Duan;R. C. Tan;Suh-Yuh Yang;Cheng-Shu You
在本文中，我们提出并分析了一种符合 $H^1$ 的混合有限元方法，用于根据电场和拉格朗日乘子求解麦克斯韦方程组，其中引入了考虑散度约束的乘子。我们主要关注物理域为非凸且其边界包含凹角或边的情况，这可能导致麦克斯韦方程组的解成为非$H^1$非常弱的函数，从而造成许多数值困难。通过向混合变分公式添加额外的依赖于网格的稳定项，将所提出的方法以稳定形式公式化。建立了稳定性和误差分析的总体框架。具体来说，研究了一对符合$H^1$的电场和乘数有限元，即$CP_2$-$P_1$元素，并推导了其稳定性和误差界。 $L$ 形域和裂纹域上的源问题和特征值问题的数值实验...
In this paper, we propose and analyze a mixed $H^1$-conforming finite element method for solving Maxwell's equations in terms of electric field and Lagrange multiplier, where the multiplier is introduced accounting for the divergence constraint. We mainly focus on the case that the physical domain is nonconvex and its boundary includes reentrant corners or edges, which may lead the solution of Maxwell's equations to be a non-$H^1$ very weak function and thus cause many numerical difficulties. The proposed method is formulated in the stabilized form by adding an additional mesh-dependent stabilization term to the mixed variational formulation. A general framework of stability and error analysis is established. Specifically, a pair of $H^1$-conforming finite elements, namely, the $CP_2$-$P_1$ elements, for electric field and multiplier is studied, and its stability and error bounds are also derived. Numerical experiments for source problems as well as eigenvalue problems on the $L$-shaped and cracked domain...