We propose a novel a posteriori error estimator for conforming finite element discretizations of two- and three-dimensional Helmholtz problems. The estimator is based on an equilibrated flux that is computed by solving patchwise mixed finite element problems. We show that the estimator is reliable up to a prefactor that tends to one with mesh refinement or with polynomial degree increase. We also derive a fully computable upper bound on the prefactor for several common settings of domains and boundary conditions. This leads to a guaranteed estimate without any assumption on the mesh size or the polynomial degree, though the obtained guaranteed bound may lead to large error overestimation. We next demonstrate that the estimator is locally efficient, robust in all regimes with respect to the polynomial degree, and asymptotically robust with respect to the wavenumber. Finally we present numerical experiments that illustrate our analysis and indicate that our theoretical results are sharp.

我们提出了一种新颖的后验误差估计器，用于二维和三维亥姆霍兹问题的有限元离散化。估算器基于平衡通量，该通量是通过解决逐点混合有限元问题而计算出的。我们表明，估计器是可靠的，直到一个因数随网格细化或多项式次数增加而趋向于一个因子为止。我们还为域和边界条件的几种常见设置推导了预因子的完全可计算上限。尽管获得的保证边界可能会导致较大的误差高估，但这会导致在不对网格大小或多项式度有任何假设的情况下进行有保证的估计。接下来，我们证明估计器是局部有效的，在多项式上的所有范围内都具有鲁棒性，关于波数的渐近鲁棒性 最后，我们提供了数值实验来说明我们的分析，并表明我们的理论结果是清晰的。