Abstract This paper is concerned with the analysis and numerical solution of an $\boldsymbol {H}({\textbf {curl}})$-elliptic hemivariational inequality (HVI). One source of the HVI is through a temporal semidiscretization of a related hyperbolic Maxwell equation problem. An equivalent minimization principle is introduced, and the solution existence and uniqueness of the $\boldsymbol {H}({\textbf {curl}})$-elliptic HVI are proved. Numerical analysis of the HVI is provided with a general Galerkin approximation, including a Céa’s inequality for convergence and error estimation. When the linear edge finite element method is employed, an optimal-order error estimate is derived under a suitable solution regularity assumption. A fully discrete scheme based on the backward Euler difference in time and a mixed finite element method in space is also analyzed, and stability estimates are derived for first-order terms of the fully discrete solution. Numerical results are reported on linear edge finite element solutions of the $\boldsymbol {H}({\textbf {curl}})$-elliptic HVI for numerical evidence of the theoretically predicted convergence order.

中文翻译：

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H (curl)-椭圆半变分不等式的数值解

摘要 本文研究了$\boldsymbol {H}({\textbf {curl}})$-椭圆半变分不等式（HVI）的分析和数值解。HVI 的一个来源是通过相关双曲麦克斯韦方程问题的时间半离散化。引入等价最小化原理，证明了$\boldsymbol {H}({\textbf {curl}})$-椭圆HVI的解存在唯一性。HVI 的数值分析提供了一般 Galerkin 近似，包括用于收敛和误差估计的 Céa 不等式。当采用线性边缘有限元法时，在合适的解规律性假设下推导出最优阶误差估计。还分析了基于后向欧拉时间差和空间混合有限元法的全离散方案，并推导了全离散解的一阶项的稳定性估计。数值结果报告了 $\boldsymbol {H}({\textbf {curl}})$-椭圆 HVI 的线性边缘有限元解，用于理论上预测的收敛顺序的数值证据。