A posteriori error estimator for N\'{e}d\'{e}lec edge element methods of two-dimensional time-harmonic Maxwell's equation
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In this paper we propose and analyze some a posteriori error estimator for the triangular N\'{e}d\'{e}lec edge element method of the two-dimensional time-harmonic Maxwell's equation. Based on the abstract result of [T. Chaumont-Frelet, arXiv:2402.17309 [math.NA] (2024)] which provides an asymptotically constant-free upper bound of Prager-Synge type, we construct the error estimator by applying elementwise minimization which seems much simpler than vertex patchwise minimizations proposed in the same reference. It is shown that the error estimator is locally equivalent to the standard residual error estimator and its local efficiency is robust with respect to jumps in the coefficients under suitable assumptions on the distributions of the coefficients. Some numerical examples are presented to support the theoretical results and illustrate the performance and accuracy of the proposed error estimator.
Abstract
In this paper we propose and analyze some a posteriori error estimator for the triangular N\'{e}d\'{e}lec edge element method of the two-dimensional time-harmonic Maxwell's equation. Based on the abstract result of [T. Chaumont-Frelet, arXiv:2402.17309 [math.NA] (2024)] which provides an asymptotically constant-free upper bound of Prager-Synge type, we construct the error estimator by applying elementwise minimization which seems much simpler than vertex patchwise minimizations proposed in the same reference. It is shown that the error estimator is locally equivalent to the standard residual error estimator and its local efficiency is robust with respect to jumps in the coefficients under suitable assumptions on the distributions of the coefficients. Some numerical examples are presented to support the theoretical results and illustrate the performance and accuracy of the proposed error estimator.