We propose a new a posteriori error estimator for mixed finite element discretizations of the curl-curl problem. This estimator relies on a Prager–Synge inequality, and therefore leads to fully guaranteed constant-free upper bounds on the error. The estimator is also locally efficient and polynomial-degree-robust. The construction is based on patch-wise divergence-constrained minimization problems, leading to a cheap embarrassingly parallel algorithm. Crucially, the estimator operates without any assumption on the topology of the domain, and unconventional arguments are required to establish the reliability estimate. Numerical examples illustrate the key theoretical results, and suggest that the estimator is suited for mesh adaptivity purposes.

中文翻译：

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旋度-旋度问题的混合有限元离散化的平衡估计器


我们提出了一种新的后验误差估计器，用于旋度-旋度问题的混合有限元离散化。该估计器依赖于 Prager-Synge 不等式，因此可以完全保证误差的无常数上限。该估计器也是局部高效且多项式鲁棒的。该构造基于分块发散约束最小化问题，从而产生一种廉价的令人尴尬的并行算法。至关重要的是，估计器的运行无需对域的拓扑进行任何假设，并且需要非常规参数来建立可靠性估计。数值示例说明了关键的理论结果，并表明该估计器适合网格自适应目的。